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GROUND-STATE CONDENSATION IN THE DILUTE HARD-SPHERE GAS
A plain-language guide for high-school readers
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by OpenAI
source: "Ground-state condensation in the dilute hard-sphere gas"
(September 24, 2026)
WHAT THIS DOCUMENT IS
---------------------
A research paper proves a theorem: in a very dilute gas of hard
spheres, when you cool it down to the lowest possible energy
level, a fixed positive fraction of the particles share the SAME
quantum state - the "constant wave" state whose wave function
does not vary in space.
We are going to break that sentence down until it means something
concrete. You will see
* what Bose-Einstein condensation is,
* why a hard-sphere gas is a good model to think about,
* why proving a fixed fraction requires new mathematics,
* and what that new mathematics is (path integrals, "bridge
replacements", "affinities", "cancellation").
You do not need to know quantum mechanics, Brownian motion, or
measure theory to start reading. Every technical term is defined
the first time it appears. Proofs are summarized "in words"
rather than expanded line-by-line, but the ideas that drive them
are given.
WHERE TO START
--------------
If you just want the punch line, read Parts 1 and 2 and stop.
If you want to see how a proof of this kind is built, read Parts
1 through 11. Every part builds on the one before.
If you are stuck on notation, flip to the GLOSSARY at the end.
CONTENTS
--------
Part 1 What is Bose-Einstein condensation?
Part 2 The mathematical setup, theorem, and plan of proof
Part 3 Ground states and their phases
Part 4 Brownian paths and the position law
Part 5 Spatial estimates
Part 6 The local rerouting machine
Part 7 Groups of labels and their conditional law
Part 8 Lattice paths through sparse obstacles
Part 9 The affinity argument (the heart of the proof)
Part 10 Recovering the phases (the geometry of pockets)
Part 11 Putting it together and taking the limit
Appendix A Glossary
Appendix B Further reading
--------------------------------------------------------------
PART 1 -- WHAT IS BOSE-EINSTEIN CONDENSATION?
--------------------------------------------------------------
1.1 THE ONE-PARAGRAPH PICTURE
-----------------------------
Particles come in two flavors. Fermions (electrons, protons, and
so on) obey the Pauli exclusion principle: no two fermions can
occupy exactly the same quantum state. Bosons (photons, helium-4
atoms, rubidium-87 atoms) are allowed - indeed encouraged - to
share states.
When you cool a gas of bosons below a critical temperature,
something dramatic happens. A macroscopic number of particles -
billions of them at once - collapse into the single lowest-
energy state available. That is Bose-Einstein condensation, or
BEC for short.
Bose and Einstein predicted it in 1924-1925 for the ideal (non-
interacting) gas. It was first seen directly in 1995, when
Cornell and Wieman cooled a gas of rubidium-87 atoms to about
170 nanokelvin and observed a spike in the velocity distribution
- a huge number of atoms all moving at the same, near-zero,
speed.
We will not measure temperatures. We will instead ask: when a
gas of interacting bosons has reached its ground state (the
state of absolute lowest energy), does a fixed positive fraction
of the particles really live in the same quantum state? That is
a purely mathematical question, and the paper we are explaining
answers YES for a specific but physically realistic model.
1.2 WHY THE QUESTION IS HARD
----------------------------
If the particles do not interact, the answer is easy. Quantum
mechanics says the ground state is the tensor product of the
lowest one-particle state with itself N times. That is dense.
When the particles do interact (say, they bounce off one
another), the ground state is no longer a simple product. The
interaction "depletes" the condensate by pushing some particles
into states above the bottom. For hard spheres, Lee, Huang, and
Yang in 1957 predicted that the depleted fraction is on the
order of the square root of the "gas parameter" - a small
number roughly equal to (density) times (sphere radius)^3.
So we know (since a 2023 paper of Fournais and Solovej, and
the 2026 upper bound of Basti and coworkers) exactly how much
energy a dilute hard-sphere gas has at ground state, to leading
order. What we do NOT know historically is whether that energy
comes from a large fraction of particles sitting in the SAME
state - the definition of condensation - or from many
different states all contributing a tiny bit.
The reason this is hard: the energy of a quantum system is
determined by a spectrum (one number), whereas condensation is
about a one-particle density matrix (a positive operator). Two
systems can have identical energy and completely different
occupation structure. Energy theorems do not imply
condensation theorems.
1.3 WHY A FIXED FRACTION, NOT A FULL ONE
----------------------------------------
The full condensate fraction for hard spheres at gas parameter
(radius)^3 * (density) is, conjecturally, 1 - O(sqrt(radius^3 *
density)). A theorem proving that would be very strong. What
this paper proves is weaker but foundational: the fraction is
at least a FIXED POSITIVE CONSTANT c_0, independent of the gas
parameter, as long as the gas is dilute enough (parameter
below some absolute threshold). That is enough to establish
that genuine BEC occurs in every ground state, irrespective of
the shape of the box or of possible degeneracies.
--------------------------------------------------------------
PART 2 -- THE MATHEMATICAL SETUP, THEOREM, AND PLAN OF PROOF
--------------------------------------------------------------
2.1 THE BOX, THE PARTICLES, AND THE HARD-SPHERE CONDITION
----------------------------------------------------------
Imagine a very large box shaped as a cube of side L. To avoid
the nuisance of walls, glue opposite faces together: whenever a
particle walks out one side, it reappears on the opposite side.
The resulting space is a "three-dimensional flat torus" and we
denote it by Lambda_L. Every point of Lambda_L is just "a
point in the box, modulo translation by L, in all three
directions".
Analogy: many video games wrap their playfield so that
walking off the right edge re-appears you on the left. That
is a 2D torus. Our box is 3D.
We put N identical bosons in Lambda_L. Physically they are
spheres of diameter a: if the center of particle i ever comes
to within distance a of the center of particle j, that
arrangement is forbidden. The set of allowed configurations is
Omega_{N,L,a}
= { (x_1, ..., x_N) in Lambda_L^N :
d_L(x_i, x_j) > a for every i < j },
an open subset of Lambda_L^N. The number a is often called the
exclusion distance or the scattering length. Since the particles
are bosons, wave functions must be symmetric: swapping any two
coordinates leaves the wave function unchanged.
2.2 WHAT A GROUND STATE IS
--------------------------
In quantum mechanics an N-particle state is a complex-valued
function Psi(x_1, ..., x_N) on the set of configurations (here,
the allowed ones), with |Psi|^2 interpreted as a probability
density. The quantity |Psi(x_1, ..., x_N)|^2 dx_1 ... dx_N is
the probability of finding particle 1 near x_1, particle 2 near
x_2, ..., particle N near x_N. The wave function is normalized
so that the total probability integrates to 1.
The kinetic-energy functional acting on Psi is
q[Psi] = sum over i=1..N of integral_{Omega} |d/dx_i Psi|^2
= integral_{Omega} |grad Psi|^2 dx_1 ... dx_N.
For hard spheres there is no potential energy: the particles
cannot come close enough to feel any force, they simply obey
the geometric exclusion. So q is the whole energy, not just
the kinetic part.
A ground state of the N-particle system is a normalized Psi
on which q attains its absolute minimum, subject to Psi = 0
on the boundary of Omega_{N,L,a} (the boundary being the set
of configurations with some pair at distance exactly a).
Analogy: imagine a rubber sheet stretched over a rocky
terrain; the lowest-energy shape of the sheet is its "ground
state". Here the "sheet" is a complex function on an N-body
configuration space, and the "energy" is the total squared
gradient.
The lowest value of q is called E_N, the ground energy.
Facts about ground states that we will use in Part 3:
* Both Psi and its modulus |Psi| are ground states with the
same energy E_N. This surprising fact is the entry point
for the "phase" argument.
* If Psi is normalized (integral of |Psi|^2 = 1) then
|Psi| is too.
* If we define Phi = |Psi| and treat the probability measure
mu(dX) = Phi(X)^2 dX
on configurations, then mu is a natural "shape" of the
ground state - all phases of Psi have been absorbed.
Later parts will treat mu as an honest probability
measure and do probabilistic computations on it.
2.3 THE CONSTANT ORBITAL AND ITS OCCUPATION
-------------------------------------------
The "constant orbital" is the wave function of a single
particle that is spread uniformly over Lambda_L:
varphi_0(x) = 1 / L^{3/2} (so integral of |varphi_0|^2
over Lambda_L is 1).
If ALL N particles were in this state, the N-particle wave
function would be the product
varphi_0(x_1) * varphi_0(x_2) * ... * varphi_0(x_N)
= L^{-3N/2},
which is a constant. A constant wave function is trivially
symmetric, so it is allowed as a bosonic state. That is what
"the whole gas condensed into the constant orbital" means.
For a general (possibly complex) ground state Psi, the
fraction of particles "in the constant orbital" is measured
by the number
B(Psi) = (1/L^3) * integral over Lambda_L^{N-1} dY of
| integral over Lambda_L dx of Psi(x, Y) |^2,
where Y = (x_2, ..., x_N) is the list of all coordinates
except the first.
That definition looks like a mouthful but has a clean meaning:
B(Psi) is the occupation of the constant orbital in the
one-particle density matrix of Psi, normalized by N.
Here is the underlying construction, in three steps:
(1) Integrate out all particles except the first, keeping
track of correlations. That produces the one-particle
density matrix Gamma^{(1)}, an operator with kernel
Gamma^{(1)}(x; y)
= N * integral over Lambda_L^{N-1} dY of
Psi(x, Y) * conj(Psi(y, Y)).
(2) The occupation of varphi_0 in Gamma^{(1)} is
N * <varphi_0, Gamma^{(1)} varphi_0>.
Because Tr Gamma^{(1)} = N, dividing by N makes this a
diagonal entry of a normalized density matrix, so it
lies between 0 and 1.
(3) That diagonal entry IS B(Psi).
An equivalent formula (used directly in the paper):
B(Psi)
= (1/L^3) * integral_{Lambda_L} dx
integral_{Lambda_L} dy
integral_{Lambda_L^{N-1}} dY
Psi(x, Y) * conj(Psi(y, Y)).
The fact that x and y appear under the SAME Y integral -
the same bath Y, not two independent ones - is the whole
reason the proof can work. The bath coordinates are what tie
"before" and "after" together.
Three facts about B to remember:
* For any normalized ground state, 0 <= B(Psi) <= 1.
* B(Psi) = 1 if and only if Psi is a constant function
(full condensation).
* B is invariant under unitary changes of length units.
Reading the box of side L with exclusion distance a as
a box of side K with exclusion distance r = aK/L, after
rescaling the wave function, preserves B. (This is a
one-line Jacobian change of variables in the double
integral above.)
That last invariance is what lets us work in "cell units" -
see Section 2.4.
2.4 SCALED UNITS: CELLS, DENSITY D, AND PARAMETER alpha
--------------------------------------------------------
The proof does almost all of its work on a scaled torus. Let
K be an integer side length, Lambda_K = (R/KZ)^3, so the torus
volume is V = K^3. Partition Lambda_K into V "unit cells"
B_v = v + [-1/2, 1/2]^3, indexed by v in (Z/KZ)^3.
Each cell is a cube of side 1; the cells tile Lambda_K.
Let D = N / V be the average number of particles per cell, and
let r be the exclusion distance in this scaled system.
Introduce the "scaled gas parameter"
alpha = D * r.
We will work in a regime where
* D is very large (the gas is dilute: many particles per
cell, but the spheres are so tiny that most cells are
mostly empty);
* alpha lies in a FIXED interval [alpha_0 / 2, 2 * alpha_0]
with alpha_0 a large absolute constant, fixed once and
for all.
Physical interpretation: since each cell has volume 1 and
contains about D particles, and each hard sphere of exclusion
distance r occupies a ball of volume (4 pi / 3) r^3, the
"gas parameter" is
(density) * (volume of one sphere)
(approximately) D * (4 pi / 3) r^3.
Now,
D * r^3 = (D * r)^3 / D^2 = (alpha^3) / D^2.
So D r^3 -> 0 as D -> infinity with alpha fixed. That is
exactly the "dilute" regime: the gas parameter shrinks as the
box grows.
The whole proof is done on Lambda_K, in units where
* D is large;
* r = alpha / D, with alpha in [alpha_0 / 2, 2 * alpha_0];
* alpha_0 is a fixed absolute constant.
At the very end (Part 11), we translate back to the original
variables N_k, L_k, a and check that the scaling is consistent.
2.5 THE MAIN THEOREM
--------------------
Here is the headline result.
THEOREM (ground-state condensation in a dilute hard-sphere gas).
There exist absolute constants eps_0 > 0 and c_0 > 0 with the
following property. Fix any density rho > 0 and exclusion
distance a > 0 satisfying
rho * a^3 < eps_0.
Consider any sequence
N_k -> infinity, L_k -> infinity,
N_k / L_k^3 -> rho.
For every choice of normalized, possibly COMPLEX, bosonic
ground eigenvector Psi_k of the N_k-particle hard-sphere
Hamiltonian on the flat torus of side L_k, its constant-
orbital occupation satisfies
liminf_{k -> infinity} B(Psi_k) >= c_0.
Read it slowly:
* Both a and rho are FIXED before the limit. Only the volume
grows. This is the thermodynamic limit at fixed density.
* The gas is dilute: rho a^3 below some threshold.
* Ground state may be complex. All ground states work.
* The lower bound c_0 is uniform over N, L, and the choice of
ground vector.
* c_0 is not given a numerical value - only its existence is
proved. It is not shown to be the correct depletion fraction
of Lee-Huang-Yang. It is just a fixed positive number.
The same c_0 works for every dilute gas parameter on the dilute
side. However, the volume threshold needed to see B >= c_0 may
depend on the density rho and on a.
What the theorem does NOT say:
* It does not identify c_0 with any explicit quantity such as
the correct condensate fraction.
* It does not show c_0 -> 1 as rho a^3 -> 0.
* It is not quantitative: it doesn't give an explicit volume
beyond which the bound kicks in.
COROLLARY (ground-space states).
Under the hypotheses of the theorem, the same bound holds for
every density operator supported on the bosonic ground space.
In particular it holds for the normalized ground-space
projection (the equal-weight mixture of all ground vectors),
and for every sequence of nonnegative ground functions.
The point of the corollary is that we don't need to select a
particular ground vector or a particular basis. Whatever the
shape of the ground space - including degenerate cases where
several complex phases coexist - a fixed fraction of every
mixture lives in the constant orbital.
2.6 HOW THE PROOF IS ORGANIZED (a roadmap)
------------------------------------------
The proof is a machine with six stages. It is worth seeing the
whole machine once before we assemble it.
STAGE A. Replace Psi by its modulus Phi = |Psi|.
This is not a loss of generality - or rather, it is a loss
we will pay for. The observation is that Psi and |Psi| are
both ground states with the same kinetic energy. Therefore
everything that holds for a nonnegative ground function
extends to a complex one, UP TO a correction controlled by a
small discarded volume. We do the whole core argument (Stages
B-D) for nonnegative Phi, then correct (Stage E) with a bound
on the discarded volume.
STAGE B. Represent Phi-path statistics via Brownian motion.
The number B(Phi) = B(|Psi|) is a double integral over
configurations (x, Y) and (y, Y). We need to explore this
double integral by moving a single particle at a time while
keeping the bath Y fixed.
The tool is a path-integral representation: there is a
probability measure R on continuous paths
omega = (omega_1, ..., omega_N) : [0, 2t] -> Lambda_K
such that each omega_i is a Brownian path (killed when any two
of them collide), and the time-slice distribution at every
time u in [0, 2t] is exactly mu(dX) = Phi(X)^2 dX. So doing
statistics on mu is the same as doing statistics on this
path measure. The path picture gives us something mu alone
does not: adjacent-in-time copies of the same particle.
The whole argument of Parts 4-6 hinges on this representation.
STAGE C. Build local rerouting moves.
Our ultimate move is "take a particle from cell u, put it in
cell v". This doesn't happen directly: instead we take a
particle whose endpoint is in cell u, and REPLACE its short
piece of trajectory with a Euclidean Brownian bridge that
starts at cell u, passes through a midpoint in cell v, and
rejoins its endpoint in cell v. This is a small tweak, but
each tweak must:
(i) keep the particle's two outer endpoints;
(ii) not collide with any other particle's trajectory;
(iii) not stray too far from its home cell.
Parts 5-6 quantify all three conditions:
-- "regularity" bounds (Part 4) ensure the replacement is
tame;
-- "collision bounds" (Part 6) ensure only a bounded
fraction of trial replacements hit other particles;
-- "supply estimates" (Parts 4-5) ensure many cells have
enough of these successful replacements to work with.
STAGE D. Average over many label choices and assemble a global
affinity gain.
Rather than fight for one tweak at a time, we:
-- partition the movable particles into disjoint groups
of size M, then handle one group at a time with all
other trajectories exposed (Part 7);
-- build two adjacent "copies" of the group's trajectories
- one where a tagged particle sits in cell u, one where
it sits in cell v - while sharing the bath Y (Part 9);
-- lower-bound their overlap (affinity) by proving a
common-measure estimate against both copies (Part 9);
-- sum over many disjoint groups and many ordered cell
pairs, multiplying by the number of admissible cell
pairs to get a fixed positive lower bound on B(Phi)
(Part 9).
This produces the core modulus bound B(Phi) >= c_* for a
fixed absolute constant c_* > 0.
STAGE E. Recover the phase loss.
When you switch from Psi to |Psi|, the two may differ in
every connected component of the allowed configuration set.
Partition the allowed slice for each fixed bath into
microscopic cells, and fill in small pockets. Any two points
that end up in what remains are connected by a path inside
one component of the allowed slice, so they share the same
phase. The volume lost by this filling-in is bounded by a
small function of r and D (Part 10). The phase-transfer
lemma then gives B(Psi) >= B(|Psi|) - 4 * sqrt(delta_geom)
with delta_geom small. Increasing D makes delta small enough
that B(Psi) >= c_*/2.
STAGE F. Take the limit and translate back.
Choose constants in a specific order (alpha_0, ell, theta,
b, t, G, M, D_0, D_*) so that all earlier estimates hold at
once. Then, given an arbitrary sequence N_k, L_k, rho, a
satisfying the dilution hypothesis, choose K_k so that the
scaled gas parameter alpha_k = D_k r_k stays in a fixed
window. All the previous machinery applies, giving B >= c_0
uniformly. This finishes the proof (Part 11).
One thing to notice about the order: c_* is fixed BEFORE
increasing D to make delta_geom small. That is essential - if
we let c_* shrink with delta_geom, we would prove nothing.
--------------------------------------------------------------
PART 3 -- GROUND STATES AND THEIR PHASES
--------------------------------------------------------------
3.1 WHAT A GROUND STATE LOOKS LIKE
----------------------------------
Recall from Part 2. A ground state of the N-particle hard-
sphere system is a normalized function Psi on the allowed
configuration set Omega_N, vanishing on the boundary (where two
particles are exactly at distance r), on which the kinetic
energy q[Psi] attains its minimum. The minimum value is E_N.
Two natural questions:
* What SHAPES can ground states have? Complex with wildly
varying phases? Or tame?
* If Psi is complex, what is the relation to its modulus
Phi = |Psi|?
The answers are beautifully restrictive, and they set up the
whole proof.
3.2 THREE FACTS ABOUT GROUND STATES
-----------------------------------
LEMMA (ground states and their moduli). For every N >= 1 with
nonempty Omega_N:
(a) The full (non-symmetric) and bosonic (symmetric) Dirichlet
operators on Omega_N have compact resolvent, and their
bottom energies coincide. So the energy E_N is genuinely
attained by an eigenfunction, not just approached.
(b) If Psi is a bosonic ground state, then its modulus
Phi = |Psi| is also a ground state (with the same E_N),
and it satisfies the same variational equation.
(c) Both Psi and Phi have smooth representatives in the
interior of Omega_N, and Phi has a definite sign on each
connected component of Omega_N. On a component where Phi
is positive, Psi = exp(i theta_C) Phi for a single
constant angle theta_C.
WHAT IT SAYS. A ground state is as tame as possible. Its modulus
is also a ground state. The complex phase of Psi can differ
between DIFFERENT connected components of the configuration
space, but cannot vary WITHIN a single component.
WHY IT MATTERS. We can do our whole calculation for a
nonnegative ground state Phi, because Phi is itself a ground
state. The cost we pay is only the phase differences between
components: we will show later that in any case the "wrong-
phase" parts are confined to a small fraction of the volume.
That is exactly the phase-transfer estimate of Section 3.5.
PROOF IDEA.
(a) Omega_N is an open bounded subset of the torus^N.
Sobolev's embedding theorem gives compactness, which makes
the Rayleigh quotient attain its minimum. The full and
bosonic energies coincide because of the "root-mean-square
symmetrization" trick:
F(X) = ( (1/N!) * sum over permutations pi of
|f(pi X)|^2 )^{1/2}.
This F is symmetric, has the same L^2 norm as f, and has
no larger kinetic energy (a Cauchy-Schwarz consequence of
pushing the gradient through the sum). So the two bottoms
coincide.
(b) Let Psi minimize the full functional. Consider Phi = |Psi|.
Since |grad |Psi|| = |Re(conj(Psi) grad Psi) / |Psi||
<= |grad Psi| almost everywhere, we have
q[Phi] <= q[Psi] = E_N.
But E_N is the minimum over all functions, so
q[Phi] >= E_N. Together: q[Phi] = E_N, so Phi minimizes too.
By the same variational principle, Phi satisfies the same
weak equation as Psi.
(c) Psi satisfies the eigenfunction equation. Standard
elliptic regularity (interior bootstrapping) gives
smoothness in the OPEN configuration set Omega_N. Since
Phi >= 0 and (-Laplacian - E_N) Phi = 0, the strong
maximum principle rules out Phi taking the value zero
inside a component without being identically zero there.
On a positive component set v = Psi / Phi. Then |v| = 1
and v is smooth. A two-line calculation shows that the
total energy difference between Psi and Phi is exactly
integral of Phi^2 * |grad v|^2.
This is nonnegative and must be zero (because the two
energies are equal). Hence grad v = 0, so v is constant on
the component.
3.3 THE MULTIPLIER IDENTITY (the most-used formula)
---------------------------------------------------
LEMMA (multiplier identity). For Phi a ground state at energy
E_N, and u any real bounded Lipschitz function on Omega_N,
q_N[u Phi] - E_N * ||u Phi||_2^2
= integral of Phi(y)^2 * |grad u(y)|^2 dy.
WHAT IT SAYS. If we "wobble" the ground state by a
multiplicative weight u, the energy increases by exactly the
integral of Phi^2 |grad u|^2.
WHY IT MATTERS. Any quantity that can be written as an
integral from Phi^2 can be bounded by choosing a suitable u
and comparing to the increased energy. Most bounds in Part 5
use exactly this. It is the tool that converts shape
information about Phi^2 into rigorous estimates.
PROOF IDEA. Phi satisfies the weak equation
q_N[Phi, w] = E_N * <Phi, w>
for every test function w. Replace w by u^2 Phi and expand.
The gradient term produces
|grad(u Phi)|^2 = u^2 |grad Phi|^2 + |grad u|^2 Phi^2
+ 2 u grad u . Phi grad Phi.
When integrated, the "2 u grad u . Phi grad Phi" term combines
with the u^2 term and cancels against E_N * <u^2 Phi, Phi>.
What is left is the stated |grad u|^2 Phi^2.
3.4 SLICES AND THE ALLOWED ONE-PARTICLE REGION
----------------------------------------------
Fix a bath configuration Y = (x_2, ..., x_N). The set of
positions x that can be added without violating the exclusion is
S_Y = Lambda_K minus the union of radius-r closed balls
centered at the particles of Y.
S_Y is called the allowed slice for bath Y. It is an open
subset of Lambda_K. A configuration (x, Y) is admissible if and
only if x lies in S_Y.
LEMMA (slices inherit the form domain). Suppose f belongs to
the form domain H_0^1(Omega_N). Then for almost every bath Y,
the function x -> f(x, Y) belongs to H_0^1(S_Y). The same holds
when any longer list of coordinates is left unfixed.
WHAT IT SAYS. Restricting a many-body wave function to a slice
at fixed bath is always a well-defined function of the
remaining coordinates, and it vanishes on the boundary of the
slice.
WHY IT MATTERS. Restricting to a slice is the only way we can
talk about individual phase components: the components of the
full Omega_N are hard to reach directly, but their intersection
with any slice is visible.
PROOF IDEA. Approximate f by smooth compactly supported
functions f_j. By Fubini, on almost every choice of the fixed
coordinates, the sequence f_j(., Y) converges in H^1(S_Y). Each
f_j(., Y) is compactly supported inside S_Y. The limit inherits
membership in H_0^1(S_Y).
COROLLARY. Two points x and y in the same connected component
of the same slice S_Y share a phase: if Psi(x, Y) and Psi(y, Y)
are both nonzero, their ratio has modulus 1.
3.5 PHASE TRANSFER: FROM |Psi| BACK TO Psi
------------------------------------------
Here is the crucial observation that lets us prove the theorem
for a complex Psi by first proving it for |Psi|. Nothing so
far tells us B(Psi) is close to B(|Psi|) - the phases between
components could be wild. We handle this by discarding a small
volume.
LEMMA (phase transfer). Suppose U_Y is a measurable subset of
allowed positions, defined for each bath Y, with the property:
for almost every Y with m(Y) > 0, the set U_Y lies inside a
SINGLE connected component of S_Y (or is empty).
Define the discarded region A_Y = Lambda_K \ U_Y and the
discarded volume fraction
delta_geom = integral of m(Y) * |A_Y| / V dY,
where |A_Y| is the Euclidean volume of A_Y and
m(Y) = integral of Phi(x, Y)^2 dx is the bath marginal weight.
(So delta_geom is the average of |A_Y|/V, weighted by the
natural bath distribution.)
Then
B(Psi) >= B(Phi) - 4 * sqrt(delta_geom).
In particular, if B(Phi) >= c_* > 0 and delta_geom <= (c_*)^2 / 64,
then B(Psi) >= c_* / 2.
WHAT IT SAYS. If we can find, per bath, a single-component
subset of the slice that covers all but a small delta_geom
fraction, then the complex occupation B(Psi) is at least the
modulus occupation B(Phi) minus a small correction.
WHY IT MATTERS. This is the pass from |Psi| to Psi. The
modulus bound proved in Part 9 (B(Phi) >= c_*) becomes a bound
on the original B(Psi), provided
we can find per-bath single-component subsets that cover
most of each slice.
Part 10 constructs these subsets: it uses small microcubes to
discard a delta_geom-small set and shows the rest sits in one
component.
PROOF IDEA. Fix a bath Y. Write
B(Phi) - B(Psi)
= (1/V) * triple integral over (x, y, Y) of
[ Phi(x,Y) Phi(y,Y) - Re(Psi(x,Y) conj(Psi(y,Y))) ].
The bracket is zero when BOTH x and y lie in U_Y (phases agree
there), and bounded by 2 Phi(x,Y) Phi(y,Y) otherwise. So
B(Phi) - B(Psi)
<= (2/V) * triple integral of
(ind_{A_Y}(x) + ind_{A_Y}(y)) Phi(x,Y) Phi(y,Y).
For the "x in A_Y" term, write the triple integral as an inner
product of two functions on (x, y, Y):
f(x,Y) = V^{-1/2} Phi(x,Y),
g(x,y,Y) = V^{-1/2} ind_{A_Y}(x) Phi(y,Y).
Cauchy-Schwarz on the (x, y, Y)-space gives
|<f, g>| <= ||f|| * ||g|| = 1 * sqrt(delta_geom).
(The f-norm is 1 because Phi is normalized and the y-integral
contributes a factor V. The g-norm is sqrt(delta_geom) by
the same computation, using m(Y) and |A_Y|.)
The "y in A_Y" term is handled identically (interchange x and
y). Summing the two terms gives the 4 in the statement.
--------------------------------------------------------------
PART 4 -- BROWNIAN PATHS AND THE POSITION LAW
--------------------------------------------------------------
4.1 THREE VIEWS OF THE MEASURE mu = Phi^2 dX
--------------------------------------------
The proof works with the "ground-state measure"
mu(dX) = Phi(X)^2 dX
on the N-particle configuration space. So far mu is just a
definition. The key move of the whole argument is a
REPRESENTATION of mu in terms of Brownian motion.
The intuition: imagine running N independent Brownian motions
for a short time 2t, KILLING the whole configuration as soon as
any two particles come within r of each other ("collision"), and
weighting every surviving path by
Phi(start positions) * Phi(end positions).
Then the weighted sum over these paths, after normalization, IS
the measure mu.
There are two equivalent ways to state this.
View (i): Heat semigroup. For the N-particle Dirichlet operator
H_N (positive, with bottom eigenvalue E_N and eigenfunction
Phi), the Feynman-Kac formula gives
(e^{-s H_N} f)(x) = E_brownian^x[ f(omega(s)) ;
killed on collision ].
By the eigenfunction equation H_N Phi = E_N Phi we get
e^{-2t H_N} Phi = e^{-2t E_N} Phi. Substituting s = 2t and
f = Phi produces the exact weighting above.
View (ii): Endpoint bridge. Fix small 2t. The joint law of
"start point y_i and end point z_i" for each particle is an
explicit weight, and CONDITIONAL on the endpoints, each
particle's path is an independent Euclidean Brownian bridge
from y_i to z_i, further conditioned on mutual avoidance.
Both views are useful. Views (i) and (ii) agree, and either can
be used as the definition.
4.2 THE STATIONARY KILLED-PATH MEASURE R
----------------------------------------
Make this precise. Let Nu_N be the measure on N independent
torus Brownian motions of generator Delta over [0, 2t],
starting from the uniform (Lebesgue) initial distribution.
Let
X(u) = (omega_1(u), ..., omega_N(u))
Y = X(0), Z = X(2t)
be the configuration at time u, its initial value, and its
terminal value.
Define the "student" weighting
R(d omega) = e^{2t E_N} * Phi(Y) * Phi(Z)
* 1_{no collision on [0, 2t]} * Nu_N(d omega).
LEMMA (stationary killed paths). R is a probability measure.
Every time marginal of R equals mu. Restricted to an interval of
length s, R has the same construction with endpoint multiplier
e^{s E_N} Phi(X_-) Phi(X_+) instead, and is invariant under
time reversal. At fixed endpoints Y and Z, R is a product of
Euclidean Brownian bridges from y_i to z_i, conditioned on
mutual avoidance.
WHAT IT SAYS. R is a genuine probability measure on paths. Its
time-marginal is mu at every time. So sampling from R at any
fixed time and sampling from mu give the same distribution.
WHY IT MATTERS. R provides temporal structure: every particle
has a trajectory, not just a position. That extra structure is
what lets us REPLACE one particle's segment with a small
"bridge" passing through a chosen cell, while keeping the other
particles' trajectories intact. mu alone has no such freedom.
PROOF IDEA. By Feynman-Kac and the eigenfunction equation,
e^{2t E_N} * <Phi, e^{-2t H_N} Phi> = 1,
so R has total mass 1 - a probability. For the marginal at
time u, split the killed kernel at u:
e^{2t E_N} (e^{-u H_N} Phi)(x) (e^{-(2t-u) H_N} Phi)(x)
= Phi(x)^2 = dmu/dx.
Endpoint disintegration: fixing Y and Z turns the two endpoint
Phi factors into constants, and leaves exactly the independent
Euclidean bridges with an avoidance indicator.
4.3 DROPPING SELECTED INTERACTIONS
----------------------------------
The interacting path measure R is a mess: N paths with mutual
avoidance. But there is a "hard-core deletion" bound stating
that, on a small set of pre-chosen labels, R behaves like the
FREE measure up to an exponential correction.
LEMMA (dropping selected interactions). Let I be a subset of
labels of size m. Let h be a nonnegative single-path functional
invariant under time reversal. Then
E_R[ prod_{i in I} h(omega_i) ]
<= e^{C t alpha m}
* integral_{mu}(dY) prod_{i in I} E^{free}_{Y_i}[h],
where E^free_{Y_i} is expectation over a free Brownian path of
length 2t starting at Y_i.
WHAT IT SAYS. The interacting expectation of a product of m
single-path functionals is at most the FREE expectation of the
same product, times the correction factor e^{Ct alpha m}.
WHY IT MATTERS. The correction is t small times alpha fixed:
exponentially close to 1. So we can extract m trajectories from
R and treat them as if they were free, at a very small cost.
All our sampling arguments on R build on this.
PROOF IDEA. Restrict to bounded h first. Fix the endpoints of
the m selected paths. Integrate the OTHER N-m paths with their
killed semigroup. The result is at most
e^{-2t E_{N-m}} * a(y) * a(z),
where a(y)^2 = integral of Phi(y, x')^2 dx' over the remaining
N-m coordinates. This is symmetric between start and end, so
the free m-path kernel weighted by a(y) a(z) has a natural
upper bound via reversal.
The energy difference E_N - E_{N-m} satisfies E_N - E_{N-m} <=
C alpha m (Part 5, Lemma sp:removal), and absorbing the
exponential gives e^{2t(E_N - E_{N-m})} <= e^{Ct alpha m}. All
that remains is the free selected-path expectation. Extend from
bounded to general h by the standard min(h, M) / monotone
convergence argument.
4.4 REGULARITY: TAMING A BROWNIAN PATH
--------------------------------------
A single Brownian path is a wild object: continuous but nowhere
differentiable, full of fractal hair. If we allow arbitrary
paths, our later estimates become unmanageable. So we impose
regularity.
Fix a mesh of the time interval [0, 2t]:
m_t = ceil(t / r^2), tau = t / m_t.
[0, 2t] is divided into 2 m_t intervals of length tau each.
For each interval I_j, define
A_j(omega) = r^{-1} * diam(tilde_omega(I_j)),
the diameter of the trajectory over that interval, measured in
units of r. Here tilde_omega is a continuous lift of omega to
Euclidean space.
Also fix three constants:
gamma = 1/12,
h_0 = D^{-1}, (the outer window length)
s_0 = b D^{-1/3}. (endpoint separation threshold)
The constant b > 0 is small and fixed later.
DEFINITION (regular path). A path omega is regular if:
(i) max over j of A_j(omega) <= D^gamma
(bounded jump per mesh);
(ii) (1/(2 m_t)) * sum over j of (1 + A_j(omega))^3 <= C_reg
(bounded average, with C_reg an absolute constant);
(iii) the outer windows [0, h_0] and [2t - h_0, 2t] have
lifted diameters at most s_0 / 8
(endpoints snug).
LEMMA (free cutoff bounds). For every fixed q > 0 and fixed
t > 0, a free Brownian path in Lambda_K fails regularity with
probability at most C_{q,t} D^{-q}. The bound is uniform in
initial point, in K, and in the allowed alpha-window.
The same bound holds for:
- a Euclidean bridge of duration 2t between endpoints at
bounded displacement;
- two Euclidean bridges of duration t joined at a fixed
midpoint, with endpoints and midpoint in a fixed bounded
region.
If, in addition, all three prescribed points lie within
distance 3 of a fixed site v, the probability that the two-leg
path leaves the radius-10 ball about v is at most C e^{-c/t}.
WHAT IT SAYS. All three conditions (i)-(iii) are extremely
likely: they lose only a polynomial factor in D and an
exponential factor in 1/t. Confinement to a fixed ball about v
is similar.
WHY IT MATTERS. Regularity is our warranty that a trial
replacement trajectory (Part 6) will:
- be confined near its home cell,
- not swing wildly across the torus,
- have its endpoint windows separated from other trajectories.
Without regularity, collision estimates fail badly.
PROOF IDEA. The increments on different mesh intervals are
Gaussian and independent. Condition (i) is a polynomial tail on
the maximum of m_t Gaussians of variance tau, using tau ~ r^2
and D large. Condition (ii) is a "2k-th moment of an empirical
average" bound via a Rosenthal / Burkholder type inequality:
each (1+A_j)^3 has uniformly bounded moments and the summands
are independent across intervals. Condition (iii) uses the
Gaussian maximum of a very short path over an interval of
length h_0 = D^{-1}, with displacement forced below s_0/8 ~
D^{-1/3}/8: probability exp(-c D^{1/3}). The confinement
bound uses the maximal Gaussian estimate for a two-leg bridge
with endpoints and midpoint all inside a small ball.
4.5 COUNTING VISITS TO A SET OF CELLS
-------------------------------------
For a set A of lattice sites, let
J_A(omega) = number of sites v in A whose radius-12 ball
the path omega visits.
J_A is an integer-valued function on paths, counting how many
prescribed cells the path "sees". Our supply estimates (Part 5,
Proposition 7) bound J_A and related sums exponentially.
LEMMA (free visits). For every lambda > 0,
E_free_y[ e^{ lambda J_A * 1_D } - 1 ]
<= zeta * e^{-d(y, A)},
where zeta = C_lambda * lambda * eps_D^{1/3},
eps_D = sup_y P_free^y[D] is the worst-case probability of the
single-path event D, and d(y, A) is the distance from y to the
nearest site of A.
WHAT IT SAYS. Under any single-path event D (however rare),
the exponential moment of visit counting is bounded above by a
"geometric" quantity that decays exponentially with distance
from A.
WHY IT MATTERS. It is our tool for "most particles do not visit
most cells". Combined with the deletion bound (Section 4.3),
it gives N-particle visit bounds without any independence
assumptions.
PROOF IDEA. Mark the lifted path at its start and at each first
exit from the unit ball about its last mark. If Q is the number
of completed unit-ball exits before time 2t, then J_A <= C(Q+1)
because the ball between successive marks can only intersect a
bounded number of radius-12 balls. The exit durations are iid
with a positive mean, so Q has every exponential moment
uniformly. If J_A > 0, the lifted path has traveled at least
d(y, A) - 12, giving a Gaussian tail in that distance.
Holder with three equal exponents, and the definitions of zeta
and eps_D, finish the estimate.
LEMMA (interacting visits). If e^{Ct alpha} * zeta <= 1, then
E_R[ exp( lambda * sum_{i=1}^N J_A(omega_i) * 1_D(omega_i) ) ]
<= exp( C e^{Ct alpha} zeta * D * |A| ).
WHAT IT SAYS. Even after summing J_A over all N particles, the
exponential moment of the total is at most exp(constant * D *
|A|), where the constant is proportional to the free per-path
"z" quantity.
WHY IT MATTERS. It converts single-path visit bounds into
N-particle visit bounds, at the cost of a factor D|A| in the
exponent.
PROOF IDEA. Expand exp(...) = prod_i (1 + h_i) with
h_i = e^{lambda J_A(omega_i) 1_D(omega_i)} - 1. Sum over
subsets I of {1, ..., N}. Apply Lemma path:deletion (Section
4.3). The free-visit lemma then bounds the result by an
exponential of e^{Ct alpha} zeta * f_A(X), where
f_A(X) = sum_i e^{-d(x_i, A)}. The spatial MGF bound for f_A
(Part 5) finishes the calculation.
4.6 CERTIFIED CELLS
-------------------
We can now formalize "a cell has many movable labels". Recall
from Part 2 that a "cell" is one of the unit cubes B_v.
For a bath Y, let n_v(Y) = number of particles of Y in B_v.
Let C_v(Y) = number of particles of Y that are BOTH within
distance 5 of v AND have a close partner: d_i(Y) <= s_0.
Define C_v(Z) similarly for the terminal configuration Z.
DEFINITION (certificate at v). A cell v has a certificate
(w.r.t. the sampled path configuration omega) if:
(i) n_v(Y) >= ell D and max(C_v(Y), C_v(Z)) < theta D,
where ell, theta are small positive constants fixed
later;
(ii) at most H_1 D trajectories visit the radius-12 ball
about v; every visiting trajectory is regular; fewer
than theta D of the visiting trajectories have lifted
diameter greater than 1.
The certificate is an event of the FULL path configuration,
not of mu alone.
PROPOSITION (supply of certificates). Fix any p_1 > 0. Choose
H_1 large, then t small, then D_0 large, so that for every set
A of distinct cells and every D >= D_0,
R( v has no certificate for every v in A )
<= p_1^{|A|}.
The bound is uniform over the chosen ground state Phi, over K,
and over alpha in the allowed window. If the parameters ell and
theta have been fixed before, then t and D_0 can be chosen
afterwards.
WHAT IT SAYS. The set of "bad" cells (no certificate) has small
probability, JOINTLY over any number of cells. Not just single
cell probability: the product p_1^{|A|} holds.
WHY IT MATTERS. It lets us select many cells and be confident
they are all usable, without ever needing independence.
PROOF IDEA. There are six failure types (one for each
inequality in the certificate). Each type individually has
probability bounded by a small parameter. Two types
(n_v(Y) and n_v(Z)) use the low-count estimate (Part 5). Two
types (C_v(Y) and C_v(Z)) use the close-count estimate (Part
5). One type bounds the count of visiting trajectories using
the interacting visit lemma with lambda fixed small. One type
bounds the count of visiting trajectories with diameter > 1,
again via the interacting visit lemma with lambda = 1 and with
D being "diameter > 1". One type bounds non-regular visitors
using the free cutoff lemma.
All six types fail with probability bounded by p_type^{|A|}.
The union-bound assignment of a single type per failing site
costs 6^{|A|} = (6)^|A|, and choosing each type's base small
enough makes the total (6 * max_type_base^{1/6})^|A| smaller
than p_1^{|A|}.
--------------------------------------------------------------
PART 5 -- SPATIAL ESTIMATES
--------------------------------------------------------------
5.1 MANY PARTICLES IN A SMALL CUBE COST ENERGY
-----------------------------------------------
One of the cleanest facts: if you force many particles into a
small region, their kinetic energy blows up. This follows from
the hard-sphere exclusion by a "ray argument".
SETTING. Let Q be a Euclidean cube of side l. Let F be a wave
function on Q^n (n coordinates, each in Q) that vanishes
wherever two particles are within r of each other. There is NO
boundary condition on the outer faces of Q.
LEMMA (cube energy). With Q and F as above,
sum over i of integral_{Q^n} |grad_i F|^2
>= c_cube * (r / l^3) * n(n-1) * integral_{Q^n} |F|^2,
with c_cube = 1 / (64 sqrt(3)).
WHAT IT SAYS. Packing n particles (exclusion distance r) into
a cube of side l costs at least (r/l^3) * n(n-1) per unit of
wave-function density.
WHY IT MATTERS. It turns local crowding into a local energy
penalty. Later, when we compare to the ground energy E_N, the
local excess is a lower bound on how far the configuration is
from equilibrium.
PROOF. The building block is the one-particle RAY BOUND:
integral_Q |f|^2
<= (sqrt(3) l^3 / r) * integral_Q |grad f|^2
for any f vanishing on a radius-r ball around some point y of Q.
To see it, consider the ray x = y + s omega. Convexity makes
the ray's intersection with Q an interval; f vanishes on
[0, r] along every ray. Then the one-dimensional inequality
|f(y + s omega)|^2
<= (1/r) * integral_r^R |d_t f|^2 t^2 dt
holds for s in [r, R], where R = R(omega) is the terminal
radius. Integrating in s and omega gives the ray bound.
Now subdivide Q into j^3 subcubes of side l/j, with
j = floor((n/2)^{1/3}). So j^3 is roughly n/2. In any sector
that assigns the n coordinates of Q^n to subcubes, at least
n/2 coordinates are NOT alone in their subcube (a counting
argument: if all but j^3 are alone, at least j^3 <= n/2
coordinates are in subcubes with partners). For each such
coordinate i, pair it with a designated subcube-mate and apply
the ray bound. This gives a lower bound
(r j^3 / (sqrt(3) l^3)) * (n/2)
times the sector norm, which is >= rn^2/(32 sqrt(3) l^3) times
the sector norm. Summing over sectors and taking the constant
gives the stated c_cube.
COROLLARY (per-cell energy). For any torus form function F and
any label subset I,
q_I[F] >= c_cube * r * sum_v n_v^I (n_v^I - 1) * |F|^2,
where n_v^I is the number of I-labels in cell B_v.
In particular the global energy satisfies
E_N / N >= c_E * alpha with c_E = c_cube / 2.
This is the crude "energy floor" in the dilute regime. It is
crude but exactly what we need to control crowding.
5.2 CLOSE PAIRS ALSO COST ENERGY
--------------------------------
A second kind of crowding involves pairs of particles that
happen to be close. The "close pair" scale here is an
intermediate length s between the exclusion distance r and the
unit cell scale 1.
LEMMA (close-neighbor energy). Let 0 < r < s < 1, let F be a
torus form function, and let I be a label subset. Define
d_i^I = min over j in I, j != i, of d(x_i, x_j)
(with value +infinity if I has no other labels). Then
q_I[F] >= c_near * r * s^{-3}
* integral of #{i in I : d_i^I <= 2s} * |F|^2,
with c_near = 27 / (262144 sqrt(3)).
WHAT IT SAYS. Every particle that has a close partner (at
distance <= 2s) costs extra kinetic energy of order r/s^3.
WHY IT MATTERS. It makes configurations with many close pairs
extremely unlikely under the ground-state measure, in a way
that does not depend on any independence assumption. It drives
the "close-count" bounds below.
PROOF IDEA. Partition each coordinate circle into intervals of
length between 8s and 16s, with a random offset. Two particles
at distance <= 2s land in the same cube with probability at
least 27/64 (an elementary volume estimate for a cube of side
between 8s and 16s). Apply the ray bound from Lemma sp:cube
inside each cube. Average over the offset. The extra factor
s^{-3} in the penalty comes from l^3 being at most 4096 s^3
and the resulting constant.
5.3 INSERTION AND REMOVAL OF PARTICLES
--------------------------------------
Energies at different particle numbers are related by inserting
a "cheap" trial orbital and comparing.
LEMMA (normalized insertion). Let w be a real normalized
smooth one-body orbital, and F an n-particle form function.
Suppose that for every configuration X in the support of F, at
most k old positions lie within distance 2r of the support of
w. If
C * ||w||_infty^2 * k * r^3 <= 1/4,
then there is an (n+1)-particle form function F~ with the same
L^2 norm as F and
q_{n+1}[F~] <= q_n[F]
+ C ( ||grad w||_2^2 + ||w||_infty^2 * k * r )
* ||F||_2^2.
WHAT IT SAYS. Inserting a particle of shape w is cheap: bounded
by a constant times (its own kinetic energy) plus a small
collision-avoidance correction.
WHY IT MATTERS. Total energies at neighboring particle numbers
differ by O(alpha) per particle. This is used constantly:
whenever we "delete" a set of labels (path:deletion, group
estimates, etc.), the correction is bounded by alpha per
deleted particle.
PROOF IDEA. Define, for each old configuration X,
f_X(y) = w(y) * min over i of h_r(d(y, x_i)),
where h_r is a nondecreasing cutoff equal to 0 up to 1.1 r,
1 beyond 2r, and Lipschitz with constant C/r. The empty
minimum is 1.
Z(X) = integral of f_X(y)^2 dy >= 3/4,
by the hypothesis on k. Define a_X(y) = f_X(y) / sqrt(Z(X)),
with sqrt(Z) replaced by max(sqrt(Z), 1/2) to keep normalization
well-defined globally.
The trial function is F(X) * a_X(y) on the (n+1)-particle
configuration space.
- The y-derivative contributes at most C ||grad w||_2^2 ||F||_2^2.
- The old-coordinate derivative of a_X is an orthogonal
projection of the gradient onto the tangent of the L^2
sphere through a_X; its squared norm is at most
||partial f_X||^2 / Z. Bounded by C ||w||_infty^2 k r ||
F ||_2^2.
- Cross terms between F and a_X vanish (the standard projection
identity: partial a_X is orthogonal to a_X in L^2(dy)).
The extra clearance 1.1 r (beyond the necessary r) and
compactly supported approximation of F show that F~ belongs to
the enlarged Dirichlet form domain.
COROLLARY (energy differences). For all sufficiently large D,
uniformly in the allowed alpha window,
E_N - E_{N-m} <= C * alpha * m for 0 <= m <= N.
Moreover, E_n / n is nondecreasing in n at fixed K and r.
WHAT IT SAYS. Deleting m particles saves at most C alpha m
energy. Energy per particle is monotone.
WHY IT MATTERS. Every deletion step in the proof costs at most
C alpha per label removed. This is the source of the e^{Ct
alpha m} factor in the path:deletion bound of Part 4.
PROOF IDEA. Use w = V^{-1/2} (the constant orbital) in Lemma
sp:insertion. The cost per insertion is at most CDr = C alpha.
Insert particles one at a time; sum the costs.
For the monotonicity, fix m < n. Average the kinetic energy of
a normalized n-particle form function over all m-label
subsets. Each coordinate is counted with probability m/n, while
the sliced energy is at least E_m by variational principle at m
particles. So E_n/n >= E_m/m. Infimum over n-particle form
functions.
5.4 LOW-COUNT BOUNDS: CELLS WITH TOO FEW PARTICLES
--------------------------------------------------
Define n_v(X) = number of particles of X in cell B_v.
PROPOSITION (low-count bound). Choose alpha_0 large, then ell > 0
small, so that for every set A of distinct cells and every
sufficiently large D,
mu{ n_v < ell D for every v in A } <= (C/D^2)^{|A|}.
The constants are independent of K and of the ground state.
WHAT IT SAYS. "Cells with too few particles" has joint probability
at most (C/D^2)^{|A|}, exponentially small in the number of cells.
WHY IT MATTERS. It justifies the "there are plenty of particles
per cell" part of the certificate (Section 4.6) - and crucially,
jointly across many cells, not just per cell.
PROOF IDEA. Choose a smooth cutoff chi_v supported inside B_v,
equal to 1 near a barycenter orbital w_v, and satisfying
|grad chi_v|^2 <= C chi_v. Let
m_v(X) = sum_i chi_v(x_i)
be a "cutoff count" in cell v (m_v <= n_v, so m_v small implies
n_v small). Let
m~_v(X) = chi_v-weighted count decayed by a Lipschitz cutoff
eta of m_v/(ell D) that equals 1 for m_v <= ell D
and 0 for m_v >= 2 ell D.
Set u_A = prod_{v in A} m~_v.
Apply the multiplier identity (Section 3.3) with u = u_A to Phi.
The left side q[u_A Phi] - E_N ||u_A Phi||^2 is bounded below by
the insertion argument: inserting ell D particles at every cell of
A costs at most (c_E alpha/2) * ell D * |A| per unit of norm^2, by
Lemma sp:insertion (use w = cw_v, a small bump at cell v). So
q[u_A Phi] - E_N ||u_A Phi||^2
>= c alpha D |A| * ||u_A Phi||^2
for a fixed small c > 0.
The multiplier identity says the SAME quantity equals the integral
of Phi^2 |grad u_A|^2. Pointwise (using disjoint supports of the
chi_v and the specific Lipschitz cutoff eta),
|grad u_A|^2 <= (C/D) sum_{v in A} u_{A \ {v}}^2.
Putting the two sides together, with a(A) = int u_A^2 dmu, gives
a(A) <= (C/(D^2 |A|)) * sum_{v in A} a(A \ {v}).
Starting from a(empty) = 1 and inducting on |A| gives
a(A) <= (C/D^2)^{|A|}. Since m_v <= n_v, the event of the
proposition forces u_A = 1, so the bound transfers.
5.5 LOCALIZED PENALTY: MASS IN A CELL COSTS ENERGY
--------------------------------------------------
This is the IMS (intermediate-value many-body step) that powers
the joint spatial bounds of Section 5.6.
Define the "exponentially weighted count" of a set A:
f_A(X) = sum_{i=1}^N e^{-d(x_i, A)},
the sum over particles of exp(-distance from that particle to
the set A of sites. When the configuration is far from A, f_A
is unusually small. When many particles crowd near A, f_A is
unusually large.
Define the "close-count near A":
k_A(X) = #{i : d(x_i, A) <= 6 and d_i(X) <= 2 s_0},
with s_0 = b D^{-1/3} and d_i = min_{j != i} d(x_i, x_j) the
nearest-neighbor distance of particle i.
LEMMA (weighted local penalty). There are constants C, c > 0
independent of b, such that every N-particle form function F
satisfies
q_N[F] - E_N ||F||^2
>= alpha * integral of
( f_A - CD|A| + c b^{-3} k_A ) |F|^2.
WHAT IT SAYS. The excess energy is at least alpha times the
integral of (weighted count near A) - alpha times a "free
volume" CD|A| + alpha times c b^{-3} times (close-mass).
WHY IT MATTERS. This is the engine behind the joint tails below.
The sign of (f_A - CD|A| + c b^{-3} k_A) controls everything:
* if f_A - CD|A| > 0, the mass near A must "pay" energy;
* if it is negative, then k_A must be at least ~ D|A|/b^3,
meaning many (near-A) close pairs, which the close-count
estimate then makes unlikely.
PROOF IDEA. Introduce a random exponential radius H (P(H >= h) =
e^{-h}). Let vartheta be a smooth profile equal to pi/2 on
(-inf, 0], 0 on [1, inf), and bounded derivative. Set
a_H(x) = sin(vartheta(d(x,A) - H))
c_H(x) = cos(vartheta(d(x,A) - H)).
So a_H^2 + c_H^2 = 1, a_H^2 is 1 for x within H, and its support
is within distance H+1. Squared gradients a_H^2, c_H^2 sum to at
most C * (indicator of transition region).
For each label subset I:
F_{I,H}(X) = F(X) * prod_{i in I} a_H(x_i)
* prod_{j not in I} c_H(x_j).
The IMS identity:
sum_I q_N[F_{I,H}]
= q_N[F] + int sum_i (|grad a_H|^2 + |grad c_H|^2) |F|^2.
So q_N[F] equals sum_I q_N[F_{I,H}] minus a localization error.
For each I:
q_{I^c}[F_{I,H}] >= E_{N-|I|} ||F_{I,H}||^2
>= (E_N - C alpha |I|) ||F_{I,H}||^2,
by the removal bound (Section 5.3).
For the I-part, split the local kinetic energy half to the cube
bound and half to the close bound:
q_I[F_{I,H}] >= (c_cube r/2) * int sum_v n_v^I (n_v^I-1) |F_{I,H}|^2
+ (c_near r/(2 s_0^3)) * int k_A |F_{I,H}|^2.
Average over H, using
int_H a_H(x)^2 <= e * e^{-d(x,A)},
int_H 1_{transition of a_H at x} <= e * e^{-d(x,A)}.
The removal costs average to at most C alpha int f_A |F|^2.
Now "leaving a positive count penalty". Within each cell,
completing the square in the count:
(c/D) n(n-1) - (C+1) e n >= -C' D
for every integer n >= 0 and D >= 1. Multiply by alpha p_v (where
p_v decays exponentially in d(v, A)) and sum over v. This leaves
the positive alpha f_A term. The overall inequality is the stated
weighted local penalty.
5.6 JOINT SPATIAL BOUNDS
------------------------
The last pieces of the spatial toolkit. Two joint bounds: an
exponential moment for the weighted count f_A, and a joint
small-probability bound for "bad" cells with too many close
pairs.
PROPOSITION (joint spatial bounds). The following hold, uniformly
in alpha in [alpha_0/2, 2 alpha_0], K >= 100, and the ground
state Phi.
(a) For every A subset L and every 0 <= z <= 1,
log integral of e^{z f_A} dmu <= CzD|A|.
Here f_empty = 0.
(b) Fix any theta > 0. Choose b small (depending on theta),
then D large. Then the random site set
Q(X) = { v in L : #{ i : d(x_i, v) <= 5 and
d_i(X) <= s_0 } >= theta D }
satisfies, for every A subset L,
mu{ A subset Q } <= p_D^{|A|},
where
p_D = (C_theta b^{-2} D^{-4/3})^{1/Q},
with Q an absolute integer (roughly the chromatic number of
the "distance <= 20" graph on lattice sites, so Q = O(1)).
As D -> infinity, p_D -> 0.
WHAT IT SAYS.
(a) The exponential moment of the weighted count f_A is
bounded by exp(C z D|A|). So mass in A, or in any set of
cells A, is only exponentially expensive (not, say,
factorial).
(b) The set of "bad" cells - those with too many close
partners nearby - has joint probability p_D^{|A|},
exponentially small in |A|. The base p_D shrinks as
D^{-4/3}.
WHY IT MATTERS. These two inequalities are the raw material for
the entire Part 6 (bridges) and Part 7 (groups). Without them
the supply estimates would need independence between cells,
which we do not have. Together they say: many cells are almost
surely simultaneously "good".
PROOF IDEA (a). Apply the multiplier identity (Section 3.3)
with u = e^{z f_A/2}. Then
q[u Phi] - E_N ||u Phi||^2
= integral of Phi^2 (z^2/4) |grad f_A|^2 e^{z f_A}.
Meanwhile the weighted local penalty (Section 5.5) says
q[u Phi] - E_N ||u Phi||^2
>= alpha * integral of (f_A - CD|A| + c b^{-3} k_A) e^{zf_A} Phi^2.
Drop the nonnegative k_A term. Also use |grad f_A|^2 <= f_A
almost everywhere (a small calculation: |grad e^{-d(x_i, A)}|^2
<= e^{-d(x_i, A)} pointwise). Let M(z) = int e^{z f_A} dmu so
M'(z) = int f_A e^{z f_A} dmu. The two inequalities become
alpha (M'(z) - CD|A| M(z)) <= (z^2/4) M'(z).
Since alpha >= 1 and z <= 1, we get M'(z)/M(z) <= C' D|A|.
Integrating from z = 0 with M(0) = 1 gives (a).
PROOF IDEA (b). Step 1: reduce to separated A. Define
m~_v(X) = sum_i a_v(x_i) * chi(d_i / s_0),
where a_v is a [0,1]-cutoff supported in the radius-6 ball about
v and equal to 1 on the radius-5 ball, and chi is a [0,1]-cutoff
equal to 1 on [0,1] and 0 on [2,inf), with |chi'|^2 <= C chi.
Then m~_v is 1 on the "close-count" set of the definition of Q,
and 0 far from it.
PACKING ESTIMATE: at almost every configuration, the gradient of
m~_v satisfies
|grad m~_v|^2 <= (C / s_0^2) * m~_v.
The reason: away from a measure-zero set, every nearest-neighbor
relation is unique. If x_i and x_j have the same nearest neighbor
x_k with |x_i - x_k| >= |x_j - x_k|, then |x_i - x_j| >= |x_i - x_k|
by the cosine law, so the two labeled directions at x_k meet at
angle >= pi/3. Packing directions on the sphere bounds the
incoming degree by a constant.
Step 2: for separated A (pairwise distance > 20 in the lattice),
define
u_A = prod_{v in A} eta(m~_v / (theta D)),
where eta in [0,1] vanishes below 1/2 and equals 1 above 1.
On the nonzero set of u_A, k_A >= theta D|A| / 2. Choose b so
that cb^{-3} theta/2 - C >= 1. Then the localized form penalty
(Section 5.5) gives
q[u_A Phi] - E_N ||u_A Phi||^2 >= alpha D |A| ||u_A Phi||^2.
At separated sites, gradients for distinct v touch disjoint
particle coordinates. So the multiplier identity plus the packing
estimate give
|grad u_A|^2 <= (C / (theta D s_0^2)) sum_v u_{A \ {v}}^2.
Let a(A) = int u_A^2 dmu. Induction from a(empty) = 1 gives
a(A) <= (C / (alpha theta D^2 s_0^2))^{|A|}
<= (C_theta b^{-2} D^{-4/3})^{|A|},
using s_0 = bD^{-1/3}.
Step 3: general A. Consider the graph on lattice sites joining
distinct sites at distance <= 20. It has bounded degree, so it
has an absolute chromatic coloring with Q colors. Any A contains
a separated color class A' of size >= |A|/Q. The event A subset Q
implies A' subset Q. Applying the separated estimate with base
at most one gives
mu{ A subset Q } <= (base)^{|A'|}
<= (base^{1/Q})^{|A|}
= p_D^{|A|}.
CHOICES. The constants are chosen in a fixed order:
alpha_0 large, then ell small;
any required theta, then b small;
finally D large.
Every threshold on D is independent of K (provided K >= 100). All
estimates hold uniformly when D varies in a compact subinterval
of its allowed range along a thermodynamic sequence.
CONCLUDING REMARK ON PART 5.
What we have so far:
* Local kinetic penalties (cube, close) that force crowding to
cost energy.
* Insertion and removal that relate E_n's at different n.
* Joint bounds saying "cells with too few particles or too
many close pairs" have exponentially small joint probability.
What we are missing: a way to CONSTRUCT, from these statistics,
successful local replacements of a particle's trajectory. That
is Part 6.
--------------------------------------------------------------
PART 6 -- THE LOCAL REROUTING MACHINE
--------------------------------------------------------------
6.0 WHAT WE WANT TO CONSTRUCT
-----------------------------
Back in Stage C of the roadmap (Part 2.6) we promised a "local
replacement" of a particle's short trajectory. Let's be precise.
Recall from Part 4 the path measure R on
omega = (omega_1, ..., omega_N) : [0, 2t] -> Lambda_K. Fix a
label i whose outer endpoints are Y_i = omega_i(0) and
Z_i = omega_i(2t).
Home cell B_v: suppose Y_i lies in cell B_v and Z_i is within
lattice distance 3 of v.
Target cells for label i: B_v itself, or one of its six face-
neighbor cells. So there are 7 possible "targets".
A TRIAL REPLACEMENT: fix a midpoint u in the target cell B.
Replace the segment omega_i on [0, 2t] by an independent pair of
Euclidean Brownian bridges:
- Bridge 1: from Y_i to u, duration t.
- Bridge 2: from u to Z_i, duration t.
- Concatenate. Project to the torus.
Call this trial SUCCESSFUL against a fixed obstacle family F if:
(a) the trial satisfies the regularity conditions of Part 4.4,
including confinement to the radius-10 ball about v;
(b) the trial stays at torus distance > r from every path in F
at every time in [0, 2t];
(c) the outer endpoints Y_i, Z_i are respected (they always
are, in this construction).
Here F is the collection of "other" trajectories - all labels
which are not being replaced currently. In the constructions of
Parts 7-9, F is a fixed collection during each comparison step.
We want to show two things:
(1) For a typical label i and any target cell B, MOST midpoint
choices u in B produce successful trials.
(2) The lucky labels are numerous: at least 9/10 of them per
certified cell (see Part 6.4).
Notice the trial keeps the OUTER endpoints Y_i, Z_i identical to
the original path. Only the middle segment changes. All estimates
below are for the trial measure
Q_{Y_i, Z_i}^B = integral over B of Q_{Y_i, Z_i}^u du,
where Q^u_{Y_i, Z_i} is the two-leg bridge law with the given
midpoint u.
The rest of Part 6 builds up to the "supply of good labels". The
key technical inputs are (6.1) a bound on the averaged density of
Brownian bridges from a packed set of endpoints, and (6.2, 6.3)
two hitting estimates against a single deterministic obstacle.
6.1 GAUSSIAN PACKING: ENDPOINTS THAT ARE SPREAD OUT
---------------------------------------------------
The first building block: when many endpoints are s_0-separated,
the density of a Brownian bridge at intermediate time is bounded
everywhere by a smooth function of that time.
LEMMA (averaged Gaussian density). Let y_1, ..., y_m in R^3
have pairwise distances at least s_0 = b D^{-1/3}, where
m >= ell D / 2. Fix u in R^3 and 0 < s <= t/2. Then
(1/m) * sum_i w_i(x)
<= C ( 1 + 1/(D s^{3/2}) )
for every x in R^3, where w_i(x) is the time-s Gaussian density
of a duration-t bridge from y_i to u (evaluated at x). The
constant C depends only on b and ell, not on t, s, D, or u.
WHAT IT SAYS. When the endpoints are s_0-separated, the AVERAGED
density of the bridge at an intermediate time s is bounded by a
quantity that behaves like 1 for s ~ D^{-2/3} (small) and is at
most C at any s.
The "1" term is the "single-endpoint" effect; the
"(1/D s^{3/2})" correction is the packing penalty from spreading
endpoints s_0 apart.
WHY IT MATTERS. Later (Proposition 6.2), the probability that a
trial hits a small obstacle is bounded by (r^3) * (averaged
density). So this bound directly controls collision probability.
PROOF IDEA. A duration-t bridge from y_i to u has time-s mean
mu_i(s) = a y_i + (1-a) u, a = 1 - s/t.
Since a >= 1/2, the means mu_i(s) are also s_0/2-separated.
The covariance is 2 s (1 - s/t) I. Within a ball of radius R
about any point, at most C(1+R/s_0)^3 of the means are enclosed.
Split the Gaussian sum into a ball of radius sqrt(s) about x
plus annuli at radii j sqrt(s), j = 1, 2, .... On each annulus,
the Gaussian decay is exp(-cj^2) uniformly. Summing:
sum_i exp(-|x - mu_i|^2/(4s))
<= C ( m / s^{3/2} * s^{3/2} + 1 * s^{3/2}/s_0^3 )
= C ( m + s^{3/2}/s_0^3 ),
after normalizing by (2 pi (s(1-s/t)) )^{3/2}. Dividing by m and
using m >= ell D/2 and s_0 = b D^{-1/3}:
1/m * (m + s^{3/2}/s_0^3) <= 1/(ell D/2) * (ell D/2 + s^{3/2} b^{-3} D)
= 1 + (2/(ell)) * s^{3/2} b^{-3}
<= C (1 + 1/(D s^{3/2})),
with the last step using 1/D in the correction: s^{3/2} <= 1
and D >= 1, so the "D" factor and the s^{3/2} factor can be
paired.
6.2 HITTING A DETERMINISTIC OBSTACLE (AVERAGED OVER LABELS)
-----------------------------------------------------------
A trial path being "successful" (Part 6.0) means avoiding every
obstacle. The complementary event is hitting a prescribed one.
For a deterministic torus path c, define
H(c) = {omega : d(omega(s), c(s)) <= r for some s in [0, 2t]}.
Let C_v be the event "the trial is regular and stays within
distance 10 of v" (from Part 4.4).
PROPOSITION (averaged collision bound). Fix m >= ell D / 2 pairs
(Y_i, Z_i) with Y_i in B_v and |Z_i - v| <= 3, and both endpoint
lists s_0-separated. Let c be a deterministic regular torus path
whose outer endpoints are at distance > s_0 from the corresponding
endpoints of every label. Then, for any target cell B,
(1/m) * sum_i Q_{Y_i, Z_i}^B ( H(c) cap C_v )
<= C t r + C r D^{3 gamma} / (D sqrt(h_0)).
If c IS one of the labels in the list, omit its own summand; the
divisor m stays.
WHAT IT SAYS. When many labels are packed s_0 apart, the AVERAGED
probability of a random trial hitting c is at most
C t r + C r D^{3 gamma} / (D sqrt(h_0)).
With gamma = 1/12, h_0 = D^{-1}, this becomes Ct r + Cr D^{-1/4}.
WHY IT MATTERS. It's the fundamental "collision cost" when a
single obstacle c is fixed, and many competing labels are
available. This is what makes the "supply" estimate of Part 6.4
work: summing over obstacles costs a factor of the count, not a
factor of the length.
PROOF IDEA. Consider the first leg from Y_i to u; time reversal
gives the identical estimate for the second leg.
Set up the time mesh: m_t = ceil(t/r^2) intervals of length
tau in [0, t]. On each interval, use the Markov property of a
Brownian bridge. If X_s is the bridge position at time s, then
on [s, s + tau],
X_{s+q} = X_s + (q/(t-s)) (u - X_s) + R_q,
where R_q = W_q - q W_{t-s} / (t-s) is a centered bridge
independent of X_s.
Two cases for s:
Case s >= h_0/2 AND s <= t/2 (early-middle). The trial is on the
"cutoff" event (|X_s - v| <= 10), so the drift term is at most
C tau / t <= r. Also, on the cutoff event, the obstacle has
diameter at most r D^gamma by regularity (it stays within
distance 12 of v). So a collision on this interval, together with
the cutoff, requires the POINT EVENT
|X_s - c-tilde(s)| <= r (2 + A_k(c) + B_s),
where c-tilde(s) is the lift of c near v, A_k is the obstacle's
mesh diameter measured in units of r, and B_s = r^{-1}
sup_{0 <= q <= tau} |R_q| satisfies E[B_s^3] <= C.
By Lemma bridge:gaussian-packing, the averaged density of X_s at
any position is at most C (1 + 1/(D s^{3/2})). B_s is independent
of X_s. By the definition of B_s, the averaged probability of the
point event is at most
C r^3 (1 + A_k)^3 (1 + 1/(D s^{3/2})).
Case s > t/2 (late): average over u is now gone, but the
conditional density of X_s at time s, given the bridge
fluctuation, is at most (t/s)^3 <= 8. Uniform in u in the
target cell. A similar point event, adding the fluctuation, gives
the same "Cr^3 (1+A_k)^3" upper bound.
Summing:
- Common "(1+A_k)^3" terms: sum_k (1+A_k)^3 <= 2 C_reg m_t
<= 4 C_reg m_t, since the obstacle is regular. So the "t r"
part sums to at most Cr t = Ctr.
- "s^{-3/2}/D" corrections: sum over s <= t/2 of s^{-3/2} is
at most C/(tau sqrt(h_0)) = C/(r^2 sqrt(h_0)). So the summed
correction is at most
C r^3 * D^{3 gamma} * D^{-1} * (r^{-2} / sqrt(h_0))
= C r D^{3 gamma} / (D sqrt(h_0)).
Combining the two sums over both legs gives the stated estimate.
Note the discipline of the split: a single sum cannot prove both
the "Ct r" part (which needs the average, which DOESN'T decay in
s) and the "D^{-1}" correction. So the argument uses the averaged
density only for short times and a conditional density for long
times. This two-part discipline is important.
6.3 SINGLE HITTING: ONE LABEL, ONE OBSTACLE
-------------------------------------------
Same setup, but no averaging over m labels - we handle ONE label
against ONE obstacle. The estimate is correspondingly stronger
in what it delivers and is used throughout Part 7.
PROPOSITION (single replacement bound). Let Y be in B_v and
|Z - v| <= 3, and let c be a deterministic regular path with
d(Y, c(0)) > s_0 and d(Z, c(2t)) > s_0. For either
Q = Q_{Y, Z}^B (any target cell B) or Q = Q_{Y, Z}^0 (the
duration-2t bridge from Y to Z directly), we have
Q(H(c) cap C_v)
<= C t r + C r D^{3 gamma} / sqrt(h_0)
<= C alpha ( t D^{-1} + D^{-1/4} ).
WHAT IT SAYS. A single trial against a single deterministic
obstacle hits with probability at most C alpha (t/D + D^{-1/4}).
This tends to 0 as D -> infinity for each fixed t.
WHY IT MATTERS. This is the "individual" comparison that appears
everywhere when we compare a proposal distribution (which uses
one trial) to a reference (which uses one trajectory): the ratio
of their densities is bounded by 1 + eta_D, with
eta_D = C alpha (t D^{-1} + D^{-1/4}). In Part 7 the quantity
eta_D is used in the compatibility estimate.
PROOF IDEA. For Q = Q^B: repeat the proof of Proposition 6.2
with one endpoint. The averaged Gaussian density is now
s^{-3/2} (only ONE label, so no packing benefit). Summing gives
the same shape.
For Q = Q^0: use the forward orientation up to time t and the
reversed orientation for the remainder. Both orientations have
duration >= t in their "remaining bridge" representation. Add
the nonnegative Ctr term.
The final inequality uses r = alpha/D and 3 gamma = 1/4:
Ctr + C r D^{3 gamma}/sqrt(h_0)
= C alpha t/D + C alpha D^{-1} * D^{1/4}
= C alpha ( t/D + D^{-1/4} ).
6.4 FROM TRIALS TO A SUPPLY OF GOOD LABELS
------------------------------------------
The pieces of Part 6 now combine.
Setting. Fix a cell v. Consider a DETERMINISTIC collection of
regular torus paths with distinct labels j, endpoints (Y_j, Z_j).
Suppose:
(H1) at most H_1 D of these paths visit the radius-12 ball
about v;
(H2) every visiting path is regular (Part 4.4).
Consider a list of m >= ell D / 2 labels satisfying:
- Y_i in B_v;
- |Z_i - v| <= 3;
- both endpoint lists are s_0-separated.
COROLLARY (supply of successful labels). For a sufficiently
small fixed t > 0 and then a sufficiently large D:
at least 9m/10 of the listed labels have the following property:
For each of the seven target cells B (home cell and its six
face-neighbors), the set
T_{i, B} = { u in B :
Q_{Y_i, Z_i}^u( successful trial ) >= 9/10 }
has spatial volume >= 99/100.
Here "successful trial" means: the trial satisfies C_v (regular
and confined to the ball of radius 10 about v) and avoids every
obstacle path in the collection except the original path of
label i itself.
All choices are uniform in the deterministic collection, in v,
and in alpha in [alpha_0/2, 2 alpha_0].
WHAT IT SAYS. Given a deterministic set of obstacles satisfying
the certificate-like conditions (H1) and (H2), at least 9/10 of
the eligible labels have: on 99/100-th-of-the-volume of EVERY
target cell, a random midpoint yields a successful trial with
probability at least 9/10.
So there is a HUGE target set T_{i, B} where "just pick u
uniformly at random" produces success with overwhelming
probability. This is the raw material for the "proposal" of
Part 7.
PROOF IDEA. Fix one listed label i. The trial can fail (relative
to the deterministic collection) in three ways:
(a) Confinement failure: the trial leaves the radius-10 ball
around v. Probability at most C e^{-c/t} (Lemma 4.4).
(b) Regularity failure: probability at most eps_D(t), which
tends to 0 as D grows, uniformly in the endpoints and
the target point.
(c) Collision with an obstacle path: only obstacles VISITING
the radius-12 ball can collide (any path not visiting that
ball is at distance > 2 from the trial, hence definitely
no collision, because the trial is confined within
distance 10 of v and r < 1).
By (H1) there are at most H_1 D visitors. Union-bounding over the
visitors and using Proposition 6.2 (averaged collision bounded by
Ctr + C r D^{3 gamma}/(D sqrt(h_0))), the integral over u in B
of the failure probability is at most
C e^{-c/t} + eps_D(t) + C H_1 D ( tr + r D^{3 gamma}/(D sqrt(h_0)) )
= C e^{-c/t} + eps_D(t) + C H_1 alpha ( t + D^{-1/4} ).
The constant multiplying t is independent of small t. Pick t
small enough that C e^{-c/t} < 1/10000 and the "C H_1 alpha t"
part is < 1/10000 (using alpha fixed). THEN pick D large enough
that eps_D(t) < 1/10000 and C H_1 alpha D^{-1/4} < 1/10000. Then
the whole integral is at most 1/70000 (for every alpha in the
allowed window).
If |T_{i, B}| < 99/100 for a fixed target cell B, then the
integral over B of the failure probability is greater than
(1/100) * (1/10) = 1/1000.
So fewer than m/70 listed labels fail the conclusion for that
specific target cell. Since there are seven target cells, fewer
than 7 * m/70 = m/10 fail for at least one of them. Every
remaining label has |T_{i, B}| >= 99/100 for ALL seven cells.
REMARK (why midpoint must be averaged). The estimate in the
corollary bounds the AVERAGE of |T_{i, B}| over the target cells
(and over the choice of midpoint), NOT a prescribed midpoint.
Indeed an estimate uniform in u would be FALSE. Example: let c be
the deterministic path that passes exactly through the midpoint
u = c(t) of the trial. Then the trial collides with c at time t
with probability 1, for EVERY midpoint equal to that special
value. So uniform-in-u estimates fail.
What we DO have: if we condition the midpoint u on a measurable
set U with volume >= a > 0, then the trial's success probability
under Q^U_{Y, Z} (the average over U) drops by at most a factor
1/a. Conditionally on an event of probability >= a, the same
holds. These two changes of measure are used in Part 7 and Part 9
to turn averages into per-cell positivity.
CONCLUDING REMARK ON PART 6.
Part 6 gives us a "supply of successful replacements" in a
precise, quantitative form:
For many labels at a certified cell, and for want of one
target cell among seven, the set of "lucky midpoints" T_{i, B}
is at least 99/100 of the cell.
What Part 6 does NOT do:
* It does not compare BEFORE (the given configuration) with
AFTER (the modified configuration). That requires a
common-measure construction (Part 9).
* It does not select many disjoint groups of labels. That is
Part 7.
* It does not move labels along a long lattice path. That is
Part 8.
* It only gives ONE label of supply per target cell at a time.
Part 7 handles many simultaneous labels.
--------------------------------------------------------------
PART 7 -- GROUPS OF LABELS AND THEIR CONDITIONAL LAW
--------------------------------------------------------------
7.1 ELIGIBLE LABELS AT A CERTIFIED CELL
---------------------------------------
Part 6 gives a "supply of successful replacements" for a single
label. But the proof needs to move MANY labels at once, along a
long lattice path (Part 8), while keeping the other N - 1
trajectories under control. Handling them all directly would
multiply collision costs. So we partition into independent small
groups and handle them one at a time.
First, eligibility.
DEFINITION (eligibility at v). A label i (its whole trajectory
omega_i, not just its endpoint) is ELIGIBLE at cell v if:
(i) Y_i = omega_i(0) is in B_v, and the final endpoint Z_i
is within distance 3 of v.
(ii) Both Y_i and Z_i are s_0-separated from all other labels:
d(Y_i, Y_j) > s_0 and d(Z_i, Z_j) > s_0 for every j != i.
(iii) The trajectory is regular (Part 4.4) and stays within
distance 10 of v on all of [0, 2t].
Two remarks:
- Eligible sets at distinct cells are automatically DISJOINT,
because the initial cell B_v of a given label is unique
(unit cells partition the torus).
- Because K is large, the radius-10 ball about v lifts
injectively to Euclidean space; so we can speak of the
"unique nearby lift" of an eligible endpoint.
PROPOSITION (certified cells have many eligible labels). Fix a
cell v with a certificate (Part 4.6). Then:
- There are at least ell D / 2 eligible labels at v.
- Of those, at least nine tenths satisfy the STRONGER property:
for every target cell B_w (home and 6 face-neighbors),
|{ u in B_w : Q_{Y_i, Z_i}^u( C_i(F_i^all) ) >= 0.9 }|
>= 0.99 * |B_w|,
where F_i^all is the full collection of all OTHER
trajectories in the sampled configuration, and |.| denotes
spatial volume within B_w (=1).
WHAT IT SAYS. A certified cell has PLENTY of eligible labels
(at least ell D / 2), and about nine tenths of those have HUGE
"successful midpoint" sets, simultaneously for every one of the
seven target cells.
PROOF IDEA. The certificate gives:
- >= ell D trajectories with Y_i in B_v (the low-count part);
- fewer than theta D with lifted diameter > 1 (from the
certificate's diameter condition);
- fewer than theta D with close partners (the close-count part);
- at most H_1 D visitors, all regular (from the certificate's
visitor bounds).
First eliminate the "bad" ones:
- lifted diameter > 1: throw away;
- close partners: throw away (both ends).
The remaining labels have Y_i in B_v, |Z_i - v| <= 2 (by the
triangle inequality), and s_0-separated endpoints. Total removed
is at most 3 theta D. Choose 3 theta < ell / 2 (this is the
"theta-choice" of Part 8 geometry). So >= (ell - 3 theta) D >=
ell D / 2 remain.
Now apply Corollary 6.4 (supply of successful labels) to this
remaining list. All hypotheses of that corollary are satisfied:
- at most H_1 D visitors (from the certificate);
- all visitors regular (from the certificate);
- endpoints s_0-separated (from the eligibility definition).
At least 9/10 of the list have the strong property for all seven
cells.
7.2 DISJOINT GROUPS AT A CERTIFIED CELL
---------------------------------------
Let M >= 1 be an integer, to be fixed LATER (after the constants
in Part 6.4).
Set
G_D = floor( ell D / (4 M) ),
the number of disjoint groups per certified cell.
At a cell with at least M G_D eligible labels, choose an ORDERED
list of M G_D distinct eligible labels uniformly without
replacement, and split it into G_D successive groups of size M
(the first group is the first M labels, the second group the next
M, etc.).
If there are fewer than M G_D eligible labels, ALL groups at that
cell are empty.
The allocation randomizations at distinct cells are INDEPENDENT
conditional on the full path sample, and depend only on the
eligible lists.
7.3 THE EXPOSED DATUM B
-----------------------
Fix one group index g in {1, ..., G_D}. Let I be the union of
this group's labels across all cells.
The EXPOSED DATUM B comprises:
- both endpoint configurations Y and Z (i.e., omega(0) and
omega(2t) for every label);
- every eligibility indicator (whether i is eligible at v for
every pair (i, v) with Y_i in B_v);
- all allocation information (which labels went to which groups
at which cells);
- the COMPLETE trajectories of all labels outside I.
Denote by F the collection of those outside trajectories - the
"frozen family".
The law of B is the pushforward: stationary path law (Part 4.2)
composed with the allocation randomizations. No certificate
event is conditioned into B. The certificate is itself a
random event, and it will be handled separately (Part 7.5).
For a label i in I at cell v, define a "reference" law
P_i(d omega_i)
= 1_{C_i(F)}(omega_i) B_i(d omega_i) / a_i,
where:
- B_i is the unconditioned Euclidean bridge of duration 2t
between the near-endpoint lifts Y_i and Z_i;
- C_i(F) is the collision-avoidance event (regular, confined to
the ball of radius 10 about v, and avoids the frozen family F);
- a_i = B_i(C_i(F)) is the normalization constant.
7.4 THE EXACT CONDITIONAL LAW
-----------------------------
PROPOSITION (exact conditional law). For almost every exposed
datum B under its own law, all the normalizers a_i and the
constant Z_I are positive, and the conditional law of the
unexposed group's trajectories is
lambda_I(d omega_I)
= 1/Z_I * 1_{all pairs within I avoid each other}
* prod_{i in I} P_i(d omega_i),
where Z_I > 0.
All reference trajectories obey their own regularity and
confinement conditions (built into P_i). Each label i in I has
at most C M other labels in the same group with which a collision
is geometrically possible, for a fixed constant C.
WHAT IT SAYS. The conditional law of the unexposed group is a
CONDITIONED PRODUCT:
product of single-label laws P_i
x indicator of "all pairs within I avoid each other".
There is no interaction with the frozen family F beyond what is
built into each P_i. The only remaining joint constraint is
mutual avoidance WITHIN I.
WHY IT MATTERS. This is where the proof becomes tractable.
Because lambda_I is a conditioned product, and each P_i is a
specific Euclidean bridge law, we can:
- compute densities of proposals relative to reference;
- bound their second moments;
- sum them against the number of labels.
The cancellation lemma of Part 9 handles exactly this product
structure.
PROOF IDEA.
Step 1 (condition on endpoints). At fixed endpoints, the free
reference law is a product of torus bridges. Remove the "shape"
weight: for a fixed endpoint pair (Y_i, Z_i), the winding weight
of the torus bridge factors out and cancels in normalization.
Step 2 (expose outside). Condition on all trajectories outside I,
so the frozen family F is now DETERMINISTIC. The conditional law
of the unexposed paths is the conditioned product:
- regularization, confinement, avoidance of F: individual
restrictions, so they factor over labels;
- mutual avoidance within I: joint restriction.
Step 3 (normalize). The integral of the conditional density
(against the product of P_i) is positive: this is the definition
of the conditional law being well-defined (which it is, because
B was sampled from an actual realization). Positivity implies
each a_i > 0 and Z_I > 0.
Step 4 (bounded interactions). A trajectory retained in I stays
within distance 10 of its home cell v_i, so it can only collide
with other labels whose home cell is within 20 of v_i. There are
at most a fixed number of cells (geometric packing constant) at
that distance, and each holds at most M labels (from the group
size). So at most C M possible colliding partners in I.
All ergodic/measure issues are handled by disintegration on the
continuous-path space: integrate against bounded functions of
the exposed and unexposed variables and apply Fubini.
7.5 ACCESSIBILITY AND SUPPLY OF GOOD CELLS
------------------------------------------
Two more definitions at the level of the exposed datum B.
DEFINITION (target sets, accessible labels).
For i in I and target cell B_w, let
T_i(w) = { u in B_w : Q_{Y_i, Z_i}^u( C_i(F) ) >= 0.9 }.
A label i is ACCESSIBLE if |T_i(w)| >= 0.99 for every one of
the seven target cells w.
For a cell v, let A_v be the set of accessible labels of the
fixed group g that happen to have Y_i in B_v. The cell v is
GOOD if |A_v| >= M/2, and BAD otherwise.
All these sets (T_i(w), A_v, the good/bad indicator of v) are
B-measurable: they depend on F, on the endpoint data, and on the
allocation, but NOT on the unexposed trajectories.
PROPOSITION (unconditional supply of good cells). For every
prescribed p > 0, the constants before M can be fixed and then M
can be chosen large enough so that, for sufficiently large D and
K, for every set A of distinct cells,
P_B( A subset {bad sites} ) <= p^{|A|}.
The bound is uniform in the group index, in the ground state Phi,
and in the allowed alpha window. If M is increased before the
lower threshold on D is raised, the conclusion remains true.
WHAT IT SAYS. Bad cells have joint probability p^{|A|}
exponentially small in |A|, JOINTLY across many cells.
WHY IT MATTERS. Part 8 moves labels along a lattice path from
cell to cell. At each visited site we need a "good" cell (so
that the move has enough accessible labels). This proposition
gives us exactly the joint small-probability of many bad cells.
PROOF IDEA. Fix the entire path sample first, before allocation.
Then apply the group allocation.
- A cell either has no certificate, or has one. No-certificate
cells: probability at most p_1^{|A|} (Part 4.6). Choose
p_1 < p/2.
- A certified cell has >= ell D / 2 >= M G_D eligible labels
and 9/10 of them satisfy the stronger property of Section
7.1. The group's M labels are drawn uniformly from this
eligible list, without replacement. For any k specified
positions, the probability that ALL k lack the stronger
property is at most 10^{-k}.
- So the probability that fewer than M/2 of the group's labels
are "strong" (i.e., the cell is bad despite having a
certificate) is at most
q_M = 2^M * 10^{-M/2},
by a union bound over choices of which k <= M/2 labels are
the "strong ones" and which are the "not strong" ones.
- Conditional on the full sample, the allocation is independent
between cells. So the allocation-failure events at distinct
certified cells are INDEPENDENT given the sample.
For each prescribed set A of sites, split A = A_0 (no
certificate) union A_1 (certified but bad). Conditional on the
sample:
P(allocation-fails on A_1) <= q_M^{|A_1|}.
Averaging over the sample and using joint certificate bounds:
P(A_0 = no-certificate set) <= p_1^{|A_0|}.
Combining:
P_B(A subset bad sites)
<= sum over A_0 subset A of p_1^{|A_0|} q_M^{|A|-|A_0|}
<= (p_1 + q_M)^{|A|}.
Choose M so large that q_M < p/2, and recall p_1 < p/2. Then
(p_1 + q_M)^{|A|} <= p^{|A|}, as claimed.
7.6 ONE-PATH COMPARISON AT FIXED EXPOSED DATA
---------------------------------------------
The next-to-last piece of Part 7. The "proposal" and "reference"
distributions differ by a bounded factor, and against a
DETERMINISTIC opponent, they collide with probability at most a
small quantity eta_D that tends to 0.
Setup. Let i be accessible, let B_w be a target cell, and let U be
a measurable subset of T_i(w) with volume at least 0.98.
Define the PROPOSAL Pi_i^U:
choose u uniformly in U;
sample Q_{Y_i, Z_i}^u;
condition on C_i(F) (the trial is regular, confined, and avoids
the frozen family F).
The "conditioning on C_i(F)" is important: we are conditioning on
SUCCESS, which is why we started from a set U where success is
likely.
PROPOSITION (density and deterministic compatibility). There is
a constant G >= 2, fixed after t is chosen, independent of D, K,
and M, such that
Pi_i^U << P_i,
0 <= d Pi_i^U / d P_i <= G,
integral of d Pi_i^U / d P_i d P_i = 1.
There is also a sequence eta_D -> 0, uniform in the exposed datum
B, with the following property.
For any fixed regular confined trajectory of a DISTINCT label
in I with its own recorded endpoints (an "opponent"), the
probability of colliding with that opponent is at most eta_D,
under EITHER P_i or Pi_i^U.
The opponent's label need not be accessible. We may take
eta_D = C_t ( t r + r D^{3 gamma} h_0^{-1/2} )
<= C_t' D^{-1/4}.
WHAT IT SAYS.
- The proposal Pi_i^U is absolutely continuous with respect to
P_i, with density bounded above by a fixed constant G.
- Collisions with individual deterministic opponents happen
with probability at most eta_D, which decays as D^{-1/4}.
WHY IT MATTERS.
- G is the "mixing constant" of the proposal; it appears as a
multiplicative factor in the affinity lower bound of Part 9.
- eta_D is the "compatibility" cost when comparing two changes,
also appearing in Part 9. Small eta_D means comparisons are
cheap.
PROOF IDEA. Let g_i be the midpoint density of B_i (the
unconditioned Euclidean bridge). Explicitly,
g_i(u) = Gaussian with mean (Y_i+Z_i)/2 and covariance t I.
On every target cell B_w, g_i has a positive lower bound c_t > 0,
because the endpoints are within distance 3 of v and every target
cell is in a bounded neighborhood.
Bound on a_i:
a_i = integral over B_i-cell B_w of
g_i(u) Q_i^u( C_i(F) ) du
>= 0.99 * 0.9 * c_t =: a_t,
> 0.
Here the 0.99 comes from |T_i(w)| >= 0.99 and 0.9 from the
success probability on T_i(w).
Density bound on the proposal: a successful trial has midpoint
u in U (so u in T_i(w)), and the density of Pi_i^U relative to
P_i is
d Pi_i^U / d P_i (trial with midpoint u)
= a_i 1_U(u) / ( |U| g_i(u) Q_i^u(C_i(F)) ).
Using:
- a_i <= 1 (a_i is a probability);
- |U| >= 0.98;
- Q_i^u(C_i(F)) >= 0.9 (since u in T_i(w));
- g_i(u) >= c_t;
the right side is at most
1 / (0.98 * 0.9 * c_t) =: G'.
Set G = max(2, G'). Both G and G' are independent of D, K, M,
and alpha.
Collision estimate against determined opponent. Fix a
regular confined trajectory c of a distinct label with recorded
endpoints. Under P_i:
P_i( collision with c )
<= (1/a_i) Q_{Y_i, Z_i}^{B}( C_v cap H(c) )
<= (1/a_t) * C(tr + rD^{3 gamma} h_0^{-1/2}),
by Prop 6.3 (single hitting). Dividing by a_t absorbs the cost.
Under Pi_i^U: integrate Prop 6.3's two-leg estimate over the
whole target cell B_w; restricting the integral to U only
decreases it. Then divide by |U| >= 0.98 and by the success
probability Q_i^u(C_i(F)) >= 0.9. Result: same shape.
The final eta_D bound uses r = alpha/D and 3 gamma = 1/4:
eta_D = C_t ( t r + r D^{3 gamma} h_0^{-1/2} )
<= C_t ( t alpha / D + alpha D^{-1/4} )
<= C_t' D^{-1/4}.
All quantities are uniform in the exposed datum B and in alpha
in the allowed window.
CONCLUDING REMARKS ON PART 7.
We have now built the "conditional-probability" machinery:
- Certified cells give >= ell D/2 eligible labels, 9/10 strong.
- Groups give G_D ~ ell D/(4M) independent sub-collections.
- For a chosen group, the exposed datum B is conditioned.
- The law of the unexposed group is an exact CONDITIONED PRODUCT.
- Bad cells are rare jointly across any set A of cells.
- Individual comparisons against deterministic opponents are
cheap (eta_D -> 0), and the proposal is bounded by G.
What Part 7 does NOT yet do:
* It does not move labels along lattice paths. That's Part 8.
* It does not actually construct the two "before/after" copies
that share the bath. That's Part 9.
--------------------------------------------------------------
PART 8 -- LATTICE PATHS THROUGH SPARSE OBSTACLES
--------------------------------------------------------------
8.0 WHY A LATTICE PATH
----------------------
Here is the shape of things so far.
Certified cells: cells with lots of eligible labels.
Groups per certified cell: G_D = floor(ell D/(4M)).
Each group has M labels. Each label can be independently
"moved" from its home cell to a face-neighbor cell (Part 6).
Now we want a version of this in the BULK: pick two distant cells
u and v, and produce a correlated pair of configurations that
differ by "shifting a label across the whole torus". Doing this
directly would multiply collision costs along the route, losing a
factor at each step. We need a different idea.
THE FIX: pick a RANDOM lattice path from u to v, along which the
moves happen ONE AT A TIME. But even this loses if we sum
"per-step costs" naively.
The actual construction (Part 9) will:
- average the change of measure over routes and labels;
- compute the SECOND MOMENT of that average;
- exploit exact cancellation of far-apart pieces, using the
encounter moment bounds provided here.
So Part 8 serves exactly one technical purpose: for a given bad
set of cells (cells with no certificate, or with too few
accessible labels), produce a "random walk path" from u to v
whose encounters with an independent copy are controlled by a
governing exponential moment. Everything else in Part 9 is
built on top of this supply.
8.1 THE LATTICE AND THE BAD SET
-------------------------------
Work on the torus lattice
L_K = (Z/KZ)^3
with its periodic maximum distance d_infty. Nearest-neighbor
edges join vertices that differ by 1 in exactly one coordinate.
Star edges (24-neighborhood) join vertices at maximum distance 1
in each coordinate. Simple paths visit each vertex at most once.
Let D be a random datum. A BAD SET is a random subset
B subset L_K. Everything else is "good". Assumption:
P( A subset B ) <= p^{|A|} for every A subset L_K.
Here p > 0 is a small parameter. This is exactly the "joint
good-cell supply" we established in Part 7.5: bad cells have
jointly small probability, exponentially in |A|.
8.2 ADMISSIBLE PAIRS
--------------------
Say an ordered pair (v, w) of distinct vertices is ADMISSIBLE
if the displacement w - v satisfies, for some integer representatives
(n, d_2, d_3):
ceil(K/10) <= n <= floor(K/8),
|d_2| <= n,
|d_3| <= n.
Here n is the FIRST coordinate displacement (picked in a large
window), and d_2, d_3 are transverse displacements (bounded by n).
So v and w are separated by about K/10 to K/8 in the first
coordinate, and up to that much sideways.
FACT (from Part 8.5). At least upsilon_0 K^6 = K^6 / 2000
admissible ordered pairs exist for K >= 80.
Reason: choose n in [K/10, K/8] (about K/40 choices), then
(2n+1)^2 transverse displacements (about K^2/25 choices), then
K^3 starting vertices. Multiply. The distinctness in the
transverse direction holds because 2n < K.
8.3 ENCOUNTERS OF TWO PATHS
---------------------------
Given two simple paths pi, pi' (each a nearest-neighbor walk on
L_K), define their ENCOUNTER COUNT
J_R(pi, pi')
= sum over z in pi
sum over z' in pi'
1_{d_infty(z, z') <= R}.
This is the number of ordered pairs of vertices, one on each
path, at lattice distance at most R. It measures "how much the
two paths chew on each other".
PROPOSITION (good paths with bounded encounters). Fix an
integer R >= 1. There are constants p_0 > 0, kappa > 0, C_path <
infinity, and an integer K_0, all depending only on R, with the
following property.
Suppose K >= K_0 and P(A subset B) <= p^{|A|} for p <= p_0.
Then for every admissible pair (v, w), there is an event
E_{v,w} of the random datum D with P(E_{v,w}) >= 1/2.
On this event, there is a measurable probability KERNEL:
a way of sampling a random simple good nearest-neighbor path
pi from v to w, and this kernel is D-measurable.
Two independent draws pi, pi' from this kernel, conditionally
on D, satisfy
E[ 1_{E_{v,w}} * E[ exp(kappa J_R(pi, pi')) | D ] ]
<= C_path.
The constants are uniform in the datum, in its law, and in the
endpoint pair.
WHAT IT SAYS. Even in a BAD environment (with bad set B), we can
find "good" paths from v to w such that two independent copies
encounter each other only "mildly" - so mildly that the
exponential moment of their encounter count is bounded by a
CONSTANT that does not grow with K.
Note the two nested expectations. The outer E averages over the
random datum D. The inner E[. | D] averages over the two path
draws GIVEN the environment. The indicator 1_{E_{v,w}} is the
"environment is not catastrophic" event. This is NOT a
statement about every fixed environment (that would be false);
it is a statement about the AVERAGED moment.
WHY IT MATTERS. In Part 9 we will bound the second moment of a
certain averaged likelihood. The bound is an exponential
function of J_R(pi, pi'), which is exactly the quantity in this
proposition. If we know E[e^{kappa J_R}] is bounded, the second
moment is bounded.
The proof of Prop 8.3 has two distinct ingredients:
(A) An explicit "skeleton" law: a random nearest-neighbor walk
from v to w whose encounter statistics we can compute,
imported from a companion paper (Section 8.4).
(B) A "detour" procedure that, on the good event, reroutes the
skeleton around finite bad components so its result is
good, without increasing encounters much (Sections 8.6-8.7).
The rest of Part 8 spells both out.
8.4 THE SKELETON LAW (imported from a companion paper)
------------------------------------------------------
The proof uses one external input: an explicit random nearest-
neighbor walk on L_K whose encounter statistics between two
independent copies are controlled. It is imported from a
companion paper and used here as a black box. Every constant in
it is a pure geometric or Fourier quantity, independent of the
gas dynamics.
CONSTRUCTION. Fix an admissible pair (v, w), lift v to R^3,
and give w the lift w + (n, d_2, d_3) with n, d_2, d_3 the
integers of Part 8.2. For each j in {2, 3}, use independent
random variables U^{(j)}_{l, k} that are uniform on [-1, 1],
indexed by integers l, k >= 0.
Define "velocity" variables and their cumulative sums:
v_i^{(j)} = c_vel * sum_{l >= 0} 2^{-l/4} U^{(j)}_{l, floor((i-1)/2^l)}
Z_t^{(j)} = sum_{i=1}^t v_i^{(j)}, Z_0^{(j)} = 0
with c_vel = (1 - 2^{-1/4})/2, so c_vel * sum_l 2^{-l/4} = 1/2.
These are sums of independent uniform variables with
exponentially decaying weights - a "Hurst-like" random
perturbation of a linear drift with index ~ 1/4 (subdiffusive at
each layer of the hierarchy). The series converges absolutely
for every sample.
Given (d_2, d_3, n), define the base vertices
u_t = v + ( t,
floor( t d_2 / n + Z^{(2)}_{min(t, n-t)} ),
floor( t d_3 / n + Z^{(3)}_{min(t, n-t)} ) )
for 0 <= t <= n.
The endpoints are u_0 = v and u_n = v + (n, d_2, d_3) = w
(up to the integer lift). Each base increment advances the
first coordinate by 1 and changes each transverse coordinate
by at most 2.
REFINEMENT. Each base increment is turned into a nearest-neighbor
path by first making the first-coordinate step, then all required
steps in coordinate 2, then coordinate 3. This takes at most 5
nearest-neighbor steps. The OCCURRENCE LIST records every vertex
of the resulting walk, without duplicating the vertex shared
between successive refinements.
Denote the projected walk by Gamma; retain this lift for the
detour construction (Section 8.7).
ADJACENT STATISTICS. For two independent copies Gamma, Gamma'
of this law and a fixed integer R_s >= 1, define the OCCURRENCE
ENCounter count
J_occ^{(R_s)} =
sum_{z in Gamma} 1_{d_infty(z, Gamma') <= R_s}
+ sum_{z' in Gamma'} 1_{d_infty(z', Gamma) <= R_s}.
Both sums count OCCURRENCES (with repetitions, because the same
vertex may appear on Gamma multiple times).
For a fixed B > 0, define the "soft-distance" statistic
S_B(Gamma, Gamma')
= sum_{z in Gamma} exp( -d_infty(z, Gamma') / B ).
COMPANION ESTIMATES (the only imports):
(IMP1) E[e^{alpha_s J_occ^{(R_s)}}] <= C_hit,
alpha_s = c_hit / (1 + R_s)^2.
(IMP2) E[e^{xi(B) S_B(Gamma, Gamma')}] <= C(B),
xi(B) > 0.
Here c_hit > 0 and C_hit < infinity are absolute, and C(B),
xi(B) depend only on B. Both bounds are uniform in K and in the
endpoint pair.
REPLACING J_occ WITH J_R. Convert the occurrence count to our
pair count:
J_R(Gamma, Gamma') <= 5 (2R + 1) J_occ^{(R)}. (pair-occ)
Reason: every first-coordinate level of a lifted skeleton has at
most 5 occurrences (level 0 has only the initial vertex; every
other level t has only vertices introduced in the t-th
refinement). The first-coordinate span is n <= K/8 < K, so the
periodic distance between distinct levels equals ordinary
distance. A fixed occurrence of Gamma can have at most 5(2R+1)
opposing occurrences within distance R.
8.5 THE EXTERIOR SHELL LEMMA
----------------------------
We need a way to route a lattice path AROUND a finite bad
cluster without breaking the "good vertices only" requirement.
The tool is the exterior boundary shell of a finite *-connected
set.
For a finite *-connected nonempty set C in Z^3, let fill(C)
consist of C together with all finite nearest-neighbor
components of Z^3 minus C. The remaining (unique) infinite
component is the exterior.
LEMMA (exterior shell). For finite *-connected nonempty C:
(a) fill(C) lies in the coordinate bounding box of C.
(b) The exterior vertices at maximum distance 1 from C form
a nearest-neighbor connected set, denoted d_*ext(C), with
at most 26|C| vertices.
(c) If C is a *-component of a larger bad set B, every vertex
of d_*ext(C) is good.
WHAT IT SAYS. Around every finite bad cluster there is a
connected "ring" of good vertices of bounded size, on which we
can walk without leaving good territory.
PROOF IDEA.
(a) A vertex outside the coordinate bounding box has a
coordinate ray to infinity that misses C, so it is not in a
finite complementary component.
(b) The shell's connectivity is Kesten's boundary-connectivity
theorem in Timar's formulation: for a finite *-connected
subset of Z^3, its exterior *-neighbors (accessible from
infinity through the nearest-neighbor complement) are
nearest-neighbor connected. The vertices in d_*ext(C) are
precisely these accessible neighbors.
There are at most 26 possible *-neighbors per vertex of C,
so |d_*ext(C)| <= 26|C|.
(c) A bad shell vertex would be *-adjacent to C, and hence
belong to the same bad component. That contradicts being
"outside" in the shell.
8.6 SMALL PERIODIC COMPONENTS AND THE REJECTION EVENT
-----------------------------------------------------
A subtlety: on the TORUS, a bad *-component could "wrap around"
K and thereby have non-finite lifts (different sections of the
same cluster meeting after the wrap). To prevent this, we reject
the environment if any bad component is too big.
Range-a adjacency: distinct torus vertices at maximum distance
at most a.
LEMMA (small periodic components). Suppose K > 2a and a range-a
component on L_K has m vertices with a m < K. Then each of its
lifts to Z^3 is a finite component projecting bijectively onto
it, and its coordinate span is at most a(m-1).
PROOF IDEA. Choose a spanning tree, lift its root, and lift
each edge by its unique displacement in [-a, a]^3. Distinct
torus vertices have distinct lifts. The difference between any
two lifted vertices, along the tree path, has maximum norm at
most a(m-1). Any additional edge's assigned endpoint difference
minus its short displacement lies in KZ^3 and has maximum norm
less than K, so it must vanish. All edges are consistent.
COUNTING BOUND. For each fixed range a, there is a constant A_a
such that the number of connected sets of m vertices containing a
specified root is at most A_a^m. Traverse a rooted spanning tree
depth-first: 2(m-1) steps from a finite set of displacements.
Combined with the bad-set assumption:
P(a range-a bad component has at least m vertices)
<= K^3 (A_a p)^m. (large-animal)
Now define the rejection:
(R1) Reject if some range-a_1 bad component has size >= m_K
= floor(K / (10 a_1)), with a_1 = R + 2.
(R2) Reject if either endpoint v or w belongs to the
projection of the fill of one of the remaining
(small) lifted bad components.
On rejection: E^c. On non-rejection: E_{v,w}.
CLAIM. P(E_{v,w}) >= 1/2 for suitable p_0, K_0.
PROOF IDEA. From (R1): P(reject) <= K^3 (A_{a_1} p)^{m_K}.
From (R2): if the fill of a lifted *-component of size m
contains a lift of an endpoint, then some root of the component
lies within maximum distance m of that endpoint. There are at
most C m^3 possible roots modulo K. Counting connected sets:
P(R2 reject) <= C sum_{m >= 1} m^3 (A_1 p)^m.
Total: P(E^c) <= K^3(A_{a_1} p)^{m_K} + C sum m^3 (A_1 p)^m.
Choose p_0 small so the series sums to at most 1/4. Then choose
K_0 large so that K^3(A_{a_1} p)^{m_K} <= 1/4 for every p <= p_0
and K >= K_0. So P(E_{v,w}) >= 1/2.
8.7 THE DETOUR AND ITS ENVIRONMENT AVERAGE
------------------------------------------
The final technical piece: convert the skeleton (which may pass
through bad vertices) into a GOOD path from v to w, and then
prove that the encounter bound survives. Since we REPLACE
vertices, we must re-bound J_R(pi, pi') for the modified paths
pi, pi' from the original J_R(Gamma, Gamma').
Step 1: the detour PROcedure. On E_{v,w}, apply the following
"reroute-around-bad-components" construction to the lifted
skeleton:
Whenever the lifted walk visits a lifted bad *-component C,
take its FIRST and LAST visits to C.
The vertex immediately BEFORE the first visit and the vertex
immediately AFTER the last visit both lie in the exterior
shell d_*ext(C). Reason: the prefix up to the first visit
avoids C, so the vertex before is a *-neighbor of C on the
exterior side; likewise the suffix after the last visit.
By Lemma 8.5(b), d_*ext(C) is nearest-neighbor connected. So
join the two shell vertices by a simple nearest-neighbor
path in the shell, and replace the intervening part of the
walk with this detour.
All added vertices are GOOD (Lemma 8.5(c)). Each replacement
removes at least one bad visit and adds none. So the process
terminates after finitely many steps.
Project the final walk to the torus and erase loops in
chronological order. Result: a SIMPLE GOOD nearest-neighbor
path from v to w. This defines a D-measurable probability
kernel on good paths, exactly as required by Proposition 8.3.
(Measurability: all refinement, shell-path, and erasure
choices use fixed orderings, so no randomness beyond the
skeleton and the datum is needed.)
Step 2: bound the new encounters. Use two independent nominal
samples Gamma, Gamma' in the same environment, and let pi, pi'
be their final simple paths. Let H range over range-a_1 bad
components on the torus, with a_1 = R+2 as before. On E_{v,w},
J_R(pi, pi')
<= J_R(Gamma, Gamma')
+ C_R * sum_H |H| * 1_{d_infty(H, Gamma) <= a_1 AND
d_infty(H, Gamma') <= a_1},
where C_R = 52 (2R + 1)^3. (detour-charge)
REASONING. Encounters where BOTH vertices are nominal are
bounded by the first term. Encounters involving nonnominal
vertices:
- If a nonnominal vertex belongs to the shell of C and
encounters a nominal vertex of the other walk, then C is
within distance R+1 <= a_1 of that other walk, and C meets
its own nominal walk. So this encounter gets charged to that
C.
- If two nonnominal vertices belong to shells of C and C', then
d_infty(C, C') <= R+2 <= a_1, so both components belong to
one range-a_1 component H meeting both nominal walks. Charge
each such encounter to one of its nonnominal vertices and to
this H.
- Bound on added vertices: at most 26|H| possible added
vertices, even if several lifted copies were used (projection
can only decrease). Each such vertex encounters at most
(2R+1)^3 vertices of the other simple path. Charging from
either path costs an additional factor 2. Total: at most
52(2R+1)^3 per unit vertex of H.
Step 3: average the environment. Fix both complete nominal
paths. Their independence from D leaves the bad-set bound
available for any prescribed collection of bad vertices.
Expand the exponential of the second term in (detour-charge) as
a product over actual range-a_1 components. Terms for DISTINCT
components have disjoint vertex sets. Sum over all disjoint
connected candidate sets, apply the bad-set bound, drop the
disjointness restriction, and use 1 + x <= e^x. This gives
E_D[1_{E_{v,w}} exp(kappa C_R sum_H |H| 1_{both-near})]
<= exp( sum over F (p e^{kappa C_R})^{|F|}
* 1_{d_infty(F, Gamma) <= a_1 AND
d_infty(F, Gamma') <= a_1} ),
where F ranges over all NONEMPTY range-a_1 connected torus sets.
The factor (e^{kappa C_R |F|} - 1) from the series expansion was
bounded above by e^{kappa C_R |F|}. So only JOINT bad-set
probabilities were used; no independence assumption on the
environment.
Step 4: convert the exponent to the soft-distance statistic.
If a contributing set F has size m, choose an occurrence
z in Gamma within distance a_1 of a root of F. There are at most
(2a_1+1)^3 root choices per z, and at most A_{a_1}^m connected
sets at each root.
A spanning-tree path in F has at most m-1 edges, each of
periodic maximum length at most a_1. Since F is also within a_1
of Gamma', the triangle inequality gives
d_infty(z, Gamma') <= 2a_1 + a_1(m-1) = a_1(m+1).
Thus the exponent in the environment bound is at most
sum over z in Gamma
sum_{m >= 1} q^m * 1_{d_infty(z, Gamma') <= a_1(m+1)},
where q = A_R p e^{kappa C_R} with A_R a fixed geometric
constant (absorbing root choices). This is a NONNEGATIVE upper
bound, so counting the same set more than once only increases
it.
REQUIRE q <= e^{-2}. Split q^m into q^{m/2} q^{m/2} and use
q^{m/2} <= e^{-m} only when d <= a_1(m+1) makes m >= d/a_1 - 1:
q^m = q^{m/2} q^{m/2}
<= q^{m/2} * e^{-m}
<= e * q^{m/2} * e^{-d/a_1}.
Summing the first factor over m >= 1:
sum_{m >= 1} q^m * 1_{d <= a_1(m+1)}
<= eta(p, kappa) * e^{-d/a_1},
with
eta(p, kappa) = e sqrt(q) / (1 - sqrt(q)) -> 0 as p -> 0.
Set B = a_1. Combining with the detour charge, the pair count
conversion, and the environment expansion:
E_D[1_E exp(kappa J_R(pi, pi')) | Gamma, Gamma']
<= exp( kappa C_R J_occ^{(R)}
+ eta(p, kappa) S_B(Gamma, Gamma') ).
Step 5: choose parameters and apply Cauchy-Schwarz. Choose
kappa > 0 with
2 kappa C_R <= alpha_R := c_hit / (1 + R)^2.
Then choose p_0 small so that:
q <= e^{-2}, 2 eta(p, kappa) <= xi(B),
and p_0 is below the endpoint-rejection p_0 of Section 8.6.
Then:
E exp(kappa C_R J_occ^{(R)} + eta S_B)
<= (E e^{2 kappa C_R J_occ^{(R)}})^{1/2}
* (E e^{2 eta S_B})^{1/2}
<= sqrt( C_hit * C(B) ) =: C_path < infinity,
where the first factor uses IMP1 (with 2 kappa C_R <= alpha_R)
and the second uses IMP2 (with 2 eta <= xi(B)).
Together with Step 4 this proves the annealed bound
E[ 1_{E_{v,w}} E[e^{kappa J_R(pi, pi')} | D] ] <= C_path.
Both draws use one common environment and independent skeleton
seeds. The event E_{v,w} keeps its original probability mass;
there is NO renormalization by conditioning on acceptance.
8.8 COUNTING ADMISSIBLE PAIRS
-----------------------------
Finally, count how many admissible ordered pairs there are.
For K >= 80, the interval [ceil(K/10), floor(K/8)] contains at
least K/40 - 1 >= K/80 integers (each is a possible first-
coordinate displacement n).
For each such n, there are (2n+1)^2 >= K^2/25 transverse
displacements (d_2, d_3). These are distinct modulo K because
2n < K.
Multiply by K^3 choices of the starting vertex v. The result:
K^3 * (K/80) * (K^2/25) = K^6 / 2000 = upsilon_0 K^6.
This gives the count in Proposition 8.3.
--------------------------------------------------------------
PART 9 -- THE AFFINITY ARGUMENT (heart of the proof)
--------------------------------------------------------------
9.0 THE PICTURE
---------------
Part 8 built "good paths" between distant cells. Part 9 uses
them.
The plan:
(1) Write B(Phi) exactly as a sum of AFFINITIES between cell-
localized measures P_{vw}, Q_{vw}.
(2) For each ordered cell pair, build a special common finite
measure nu_hat (the "before/after bridge") supported on a
large-probability event, and lower-bound the affinity of
P_{vw} and Q_{vw} via a common-measure estimate.
(3) Do the lower bound for many disjoint groups and average.
(4) Sum over admissible pairs using the lattice paths.
The payoff is a fixed positive lower bound B(Phi) >= c_*.
9.1 AFFINITY: THE BASIC TOOL
----------------------------
DEFINITION (affinity). For finite nonnegative measures P and Q
on the same measurable space,
H(P, Q) = integral of sqrt( p * q ) dxi,
where xi = P + Q is any common dominating measure, and p = dP/dxi,
q = dQ/dxi are their densities with respect to xi.
Using another dominating measure gives the same value (this is a
standard fact about Radon-Nikodym derivatives). If P, Q are
BOTH probabilities and Q << P with density dQ/dP bounded, then
H(P, Q) is the square-root overlap of their densities - a
number in [0, 1] measuring how similar they are.
Three elementary properties:
(a) MONOTONICITY. If P >= P' and Q >= Q' as measures, then
H(P, Q) >= H(P', Q').
(b) SUPERADDITIVITY. For finite lists of measure pairs,
H( sum_j P_j, sum_j Q_j ) >= sum_j H(P_j, Q_j).
Proof: pointwise Cauchy-Schwarz gives
sum_j sqrt(p_j q_j) <= sqrt((sum_j p_j)(sum_j q_j)).
(c) KERNEL CONTRACTION. Applying the SAME probability kernel
to both P and Q cannot decrease their affinity. Proof:
joint measure + conditional Cauchy-Schwarz.
LEMMA (a common finite measure; the Holder trick). Let P, Q be
finite nonnegative measures and let nu be a finite measure of
mass m > 0 that is absolutely continuous with respect to both P
and Q. If
integral of (dnu/dP)^2 dP <= C_P,
integral of (dnu/dQ)^2 dQ <= C_Q,
then
H(P, Q) >= m^2 / sqrt(C_P C_Q).
WHAT IT SAYS. If a small "common witness" nu exists inside both
P and Q with well-controlled likelihoods, then P and Q must have
substantial overlap.
WHY IT MATTERS. This is how the whole affinity lower bound will
be proved: we construct nu = nu_hat explicitly, then check that
its squared "reverse likelihoods" against P_{vw} and Q_{vw} are
bounded by a constant, and conclude H >= 1/(4 C_2) per group.
PROOF. With densities p, q, s relative to a common dominating
measure xi,
m = integral of s dxi
= integral of (s^2/p)^{1/4} (s^2/q)^{1/4} (pq)^{1/4} dxi.
Apply Holder with exponents (4, 4, 2):
m <= (int s^2/p)^{1/4} (int s^2/q)^{1/4} (int sqrt(pq))^{1/2}
= C_P^{1/4} C_Q^{1/4} H(P,Q)^{1/2}.
Squaring and rearranging gives m^2 <= C_P^{1/2} C_Q^{1/2} H, so
H >= m^2 / sqrt(C_P C_Q).
9.2 CELL MEASURES AND THE OCCUPATION FORMULA
--------------------------------------------
Recall from Part 2.3:
B(Phi) = (1/V) * int over x in Lambda_K
int over y in Lambda_K
int over Y in Lambda_K^{N-1}
Phi(x, Y) Phi(y, Y) dx dy dY.
The KEY feature: the SAME Y appears in both factors. So we are
comparing "wave function at (x, Y)" with "wave function at (y, Y)"
- two configurations that SHARE a bath Y but differ in one
highlighted coordinate.
Partition the pair (x, y) in Lambda_K x Lambda_K into ordered
cell pairs (B_v, B_w). Define, for each ordered pair (v, w):
P_{vw}(dx dy dY) = V * Phi(x, Y)^2 dx dy dY,
supported on x in B_v, y in B_w, Y in Lambda_K^{N-1}.
Q_{vw}(dx dy dY) = V * Phi(y, Y)^2 dx dy dY,
same support.
Both are finite nonnegative measures. Note the ASYMMETRY: P
weights Phi(x, Y)^2 (a "position distribution" of the first
coordinate x), Q weights Phi(y, Y)^2 (a "position distribution"
of the second coordinate y). Both integrate Y in the same way.
The label "first coordinate" is our TAGGED particle; the bath Y
is common.
PROPOSITION (cell-affinity; the occupation formula). With P_{vw}
and Q_{vw} defined as above,
B(Phi) = (1/V^2) * sum_{v, w in L_K} H(P_{vw}, Q_{vw}).
WHAT IT SAYS. B(Phi) is the average, over ALL ordered cell pairs
(v, w), of the affinity between the two measures. Adding up cell-
pair affinities reconstructs the total occupation.
WHY IT MATTERS. To LOWER-bound B(Phi), it suffices to lower-bound
the sum of the cell-pair affinities. The strategy of the rest of
Part 9:
(i) lower-bound H(P_{vw}, Q_{vw}) for ONE ordered cell pair;
(ii) show that the lower bound holds for a positive fraction
upsilon_0 of cell pairs;
(iii) sum to get B(Phi) >= upsilon_0 c.
PROOF. On a single cell pair, the affinity is
H(P_{vw}, Q_{vw})
= integral over B_v x B_w x Lambda_K^{N-1} of
sqrt( V^2 Phi(x,Y)^2 Phi(y,Y)^2 ) dx dy dY
= integral of V * Phi(x,Y) * Phi(y,Y) dx dy dY,
using Phi >= 0 so the square root is just V * Phi(x,Y) * Phi(y,Y).
Summing over all V^2 ordered cell pairs:
sum_{v,w} H(P_{vw}, Q_{vw})
= integral over Lambda_K^2 x Lambda_K^{N-1} of
V * Phi(x,Y) Phi(y,Y) dx dy dY
= V * [ integrand of B(Phi) ]
= V^2 * B(Phi).
Divide by V^2 to get the formula.
So B(Phi) is exactly a "V^2-scaled sum of cell-pair affinities",
and we must lower-bound the affinities.
9.3 TAG NORMALIZATION: BREAKING INTO GROUPS
-------------------------------------------
Cell-pair affinities H(P_{vw}, Q_{vw}) are still global objects.
We must break them into group-wise measures that retain B
(the exposed datum) and the tag.
Setting. Fix a group index g in {1, ..., G_D}. Let I_g be the
union of this group's labels across all cells. Recall (Part 7.3)
that B contains endpoints Y and Z, eligibility indicators, the
allocation, and the trajectories OUTSIDE I_g.
Define group-wise measures Phat_g and Qhat_g that retain B and
the triple (x, y, Y), as follows.
For Phat_g:
- sample B under its law;
- sample the conditional unexposed paths omega_{I_g}
according to lambda_{I_g} (Part 7.4);
- SUM over i in I_g the following contribution:
take x = omega_i(t), the tagged midpoint of omega_i;
require x in B_v;
order the OTHER N-1 midpoints by a fresh uniform
permutation of the remaining labels to form bath Y;
sample y UNIFORMLY in B_w.
For Qhat_g:
- sample B under its law;
- sample conditional unexposed paths omega_{I_g};
- SUM over i in I_g: take y = omega_i(t), tagged midpoint;
require y in B_w;
form bath Y from the OTHER N-1 midpoints by a fresh
uniform permutation;
sample the EXTRA point x uniformly in B_v.
Note the discipline: we SUM over i (not uniformly pick one). This
keeps a manageable normalization.
LEMMA (tag normalization; cmp:tag-normalization). For each
ordered pair (v, w) of cells,
H(P_{vw}, Q_{vw})
>= (1/D) * sum_{g=1}^{G_D} H(Phat_g, Qhat_g).
WHAT IT SAYS. The FULL cell-pair affinity dominates (1/D) times
the sum of group-wise affinities. The factor 1/D pays for having
to normalize the tagged particle.
WHY IT MATTERS. To lower-bound H(P_{vw}, Q_{vw}), it suffices to
lower-bound the group sum. Each group is handled by the
cancellation lemma below.
PROOF. Put all groups on the ORIGINAL stationary sample, with
all allocation randomizations and a fresh uniform permutation of
the full label set. After deleting a tag, the restricted
permutation uniformly orders its bath. Each group's conditional
construction is a disintegration of this common sample with
respect to its OWN exposed datum.
For every sampled path configuration, the group label sets
I_1, ..., I_{G_D} are DISJOINT. Their sum of tag contributions
is bounded by the sum over all N label positions. The
unrestricted midpoint vector has the symmetric density
Phi(x, Y)^2 (i.e. mu). For each FIXED tag label, uniform
ordering of the bath gives the SAME density Phi(x, Y)^2.
Summing over all tags gives N times that density.
Since the extra endpoint cell B_w has volume 1 and V/N = 1/D:
(1/D) * sum_g proj_X(Phat_g) <= P_{vw},
(1/D) * sum_g proj_X(Qhat_g) <= Q_{vw},
where proj_X means "project to drop the exposed datum B".
Now use the properties established in Section 9.1:
- MONOTONICITY: H only increases passing to bigger measures.
- KERNEL CONTRACTION: projection is a probability kernel, so
H(proj P, proj Q) <= H(P, Q).
- SUPERADDITIVITY: H of a sum >= sum of H's.
- HOMOGENEITY: H(cP, cQ) = c H(P, Q) for c > 0.
Putting these together:
H(P_{vw}, Q_{vw})
>= H( (1/D) sum proj Phat_g, (1/D) sum proj Qhat_g )
= (1/D) H( sum proj Phat_g, sum proj Qhat_g )
>= (1/D) sum_g H( proj Phat_g, proj Qhat_g )
>= (1/D) sum_g H( Phat_g, Qhat_g ).
9.4 THE TWO MIDPOINT ASSIGNMENTS (the common measure)
-----------------------------------------------------
The heart of the construction. Fix an admissible pair (v, w)
and R = 30. Let E_{vw} be the B-measurable event from Prop 8.3,
with P(E_{vw}) >= 1/2. (We will use the SAME event for every
group g; the group is fixed afterwards.)
Choose the group size M to satisfy the encounter condition below
(Section 9.7).
On E_{vw}, draw a measurable simple nearest-neighbor GOOD path
z_0 = v, z_1, ..., z_d = w
through good sites. The kernel above provides this.
For each k < d, INDEPENDENTLY choose a uniform ACCESSIBLE label
i_k anchored at z_k. (Anchored at z_k means its home cell is z_k;
accessible means |T_i(w)| >= 0.99 for all seven target cells w.)
These labels are distinct automatically, since their anchors
z_0, ..., z_{d-1} are distinct.
We will now construct TWO "pictures" - two probability models on
the unexposed trajectories - whose shared data is (x, y, Y).
The pictures differ in WHICH label starts where, but share the
same bath.
FIRST PICTURE: labels i_0 at U_0, i_1 at U_1, ..., i_{d-1} at U_{d-1},
extra point at U_d.
SECOND PICTURE: labels i_0 at U_1, i_1 at U_2, ..., i_{d-1} at U_d,
extra point at U_0.
U_0 U_1 U_2 ... U_d
First picture: i_0 i_1 i_2 ... extra
Second picture: extra i_0 i_1 ... i_{d-1}
So label i_k sits at U_k in the first picture but at U_{k+1} in
the second. Each label's OUTER ENDPOINTS are its own original
endpoints Y_{i_k}, Z_{i_k} - the SAME in both pictures.
CONSTRUCTION. First sample all UNSELECTED unexposed trajectories
from their exact marginal under lambda_I.
For each vertex z_k, k = 0, ..., d, prepare a factor as follows.
The first picture needs label i_k through z_k (for k < d
only). The second picture needs label i_{k-1} through z_k
(for k > 0 only).
Let T1_k = T_{i_k}(z_k) if k < d, else unconstrained.
Let T2_k = T_{i_{k-1}}(z_k) if k > 0, else unconstrained.
The set A_k := T1_k cap T2_k (only intersecting the one
required set at each endpoint z_0 or z_d). By the accessibility
property, |A_k| >= 0.98 (this is where the 0.99 with a 0.02
safety margin comes from).
Sample U_k uniformly in A_k.
For each of the one or two REQUIRED bridges through z_k (the
bridge for the first picture's i_k and/or the second picture's
i_{k-1}), sample a two-leg trial through U_k, CONDITIONED on
its individual SUCCESS against the frozen (unselected) paths.
All d+1 vertex factors are INDEPENDENT before the final
conditioning below.
USE U_d as the extra point in the first picture; U_0 as the
extra point in the second picture.
CONDITIONING STEP. Now impose all REMAINING hard-avoidance
checks in both pictures: the selected paths must avoid the
unselected paths, and one another within their own picture.
Condition the product of vertex factors on these checks
SEPARATELY for each FIXED collection of unselected paths. So the
unselected-path marginal remains its original one under lambda_I.
WELL-DEFINEDNESS CHECK. Each vertex factor carries at most two
bridges (one per picture, at most). Each bridge has at most
C M possible "opponents" in the two pictures (a fixed geometric
constant times M - because labels are confined to tubes of radius
10 about their home sites, and there are at most M labels per
site). All opponents have distinct labels within the picture
where a check is imposed.
By Prop 7.5 (density + compatibility), the deterministic
opponent comparator gives at most eta_D collision probability
per check. Summing over the <= CM opponents per bridge:
chi_D = C_col * M * eta_D -> 0
as D -> infinity, with M fixed. So each vertex factor passes
every incident check with probability >= 1 - chi_D.
Inserting the d+1 factors SUCCESSIVELY:
P(all checks pass) >= (1 - chi_D)^{d+1} > 0.
So the conditional construction is well-defined.
TAGGING. Now tag label i_0 in the first picture and i_{d-1} in
the second picture. In each picture, the tag plus the extra point
together define x = U_0 and y = U_d. The intermediate
midpoints in both pictures are exactly U_1, ..., U_{d-1}; the
unselected midpoints also agree (we conditioned on them being
identical). Choosing a common uniform ordering of the shared
midpoints gives the uniform ordering law for the bath.
This defines a SINGLE finite measure nu_hat on (B, x, y, Y), with
zero mass off E_{vw}. Each conditional sampling law on E_{vw}
is a probability measure, so:
nu_hat(all) = P(E_{vw}) >= 1/2.
So nu_hat is a common witness of mass at least 1/2.
The rest of Part 9 (Sections 9.5-9.9) checks that:
(i) nu_hat has bounded squared likelihood against Phat_g: that
is, int (d nu_hat/d Phat_g)^2 d Phat_g <= C_2, with C_2
independent of the group index and the cell pair;
(ii) the SAME bound holds against Qhat_g.
Then the Holder lemma (Section 9.1) gives the affinity bound per
group.
9.5 THE CANCELLATION LEMMA
--------------------------
Fix exposed data B with a feasible joint pattern. On this fixed
datum, the group's joint law under lambda_I is
lambda_I = (1/Z_I) * 1_{all pairwise avoidance}
* prod_{i in I_g} P_i (d omega_i),
by Prop 7.4 (Part 7.4). This is the CONSTRAINED PRODUCT form.
Our nu-hat, on E_vw, is built differently: selected labels sampled
from a conditioned two-leg bridge through a midpoint, not from
the single-path reference P_i with individual permits. But by
design (Part 7.6), the constrained bridge is absolutely
continuous with respect to the reference, with bounded density.
-----------------
CONSTRAINED PRODUCT DEFINITION. For level k, define the
CONSTRAINED PRODUCT
P_g^{A_k} (d omega_I | B)
= (1/Z_g^{A_k}) * 1_{all pairwise avoidance}
* prod_{i in I_g, selected} Pi_i^{A_k} (d omega_i)
* prod_{j in I_g, unselected} P_j (d omega_j)
* 1_{each unselected j has an individual permit at level k},
where A_k refers to the vertex-level k (Part 9.4).
Q_g^{A_k} is defined identically with Q in place of P.
LEMMA (cancellation; cmp:cancellation). For each level k and
every measurable set A_Y contained in the admissible positions
of a fixed class Y (a fixed record of which sites each exposed
label occupies):
integral_{A_Y} (d nu-hat|_{E_vw} / d P_g^{A_k}) d P_g^{A_k}
<= G * C_path * 1_E_vw * (measurable bounded factor),
and the SAME inequality holds with Q_g^{A_k} in place of
P_g^{A_k}. Both sides are zero off E_vw.
WHAT IT SAYS. nu-hat, restricted to E_vw, is dominated by BOTH the
constrained product against P and the constrained product against
Q, with a bound that involves only ONE factor G (from the
single-path comparison at Part 7.6) and the imported C_path (from
Part 8.3).
WHY IT MATTERS. Together with the Holder lemma (Section 9.1),
this bounds the per-group affinity from below by a fixed
positive number, independent of D and K.
PROOF IDEA. Both P_g^{A_k} and Q_g^{A_k} are built by drawing
each of the group's labels from its own single-path reference.
nu-hat draws the same labels from a conditioned bridge instead.
By the density bound (Part 7.6), the constrained bridge has
density at most G times the single-path reference.
Three ingredients make the bound survive as ONE G rather than
G^M, even though the group has M labels:
(i) SYMMETRY OF INTERIOR LABELS. All M labels are treated
identically. Under P_g^{A_k}, each interior label
contributes a factor with L^2 norm 1 (it is a probability
density), and these factors are independent. Their
M-fold average of squares stays bounded by 1, NOT by G^M.
(ii) THE BOUNDARY AT U_0 OR U_d. The single tag difference
between picture 1 and picture 2 occurs at one vertex (U_0
for the first picture, U_d for the second). That vertex
contributes the ONE factor G.
(iii) THE EXTRA POINT. At picture 1's own extra point (U_d) or
picture 2's (U_0), a uniform distribution with density
1/|A_{vertex}| is inserted. Since |A_{vertex}| >= 0.98,
the factor is at most 1/0.98 < 2. Squaring gives (1/0.98)^2.
Altogether, the ratio is at most
G * (1/0.98)^2 * (absorb C_path from Part 8.3 moments).
Enlarging G once and for all to absorb (1/0.98)^2 and any
remaining numerical factors (e.g. choosing G = max(2, 1/(0.98 *
0.9 * c_t) * 1.05) at Part 7.6), we record the cleaner bound
integral_{A_Y} (d nu-hat|_{E_vw} / d P_g^{A_k})^2 d P_g^{A_k}
<= C_2,
with
C_2 = G * C_path,
a fixed constant independent of D, K, group index g, ordered cell
pair (v, w), and alpha in its window.
-----------------
9.6 APPLYING HOLDER: THE PER-GROUP AFFINITY
-------------------------------------------
We now use the common-measure construction (Section 9.4) together
with the cancellation bound (Section 9.5) to lower-bound the
per-group affinity.
SETUP. Recall from Section 9.4:
- nu-hat is a finite measure on (B, x, y, Y), with
nu-hat(all) = P(E_vw) >= 1/2.
- nu_hat << Phat_g and nu_hat << Qhat_g.
- The squared-likelihood bounds hold:
int (d nu_hat / d Phat_g)^2 d Phat_g <= C_2,
int (d nu_hat / d Qhat_g)^2 d Qhat_g <= C_2.
The second of these is never directly verified: what we verify in
Section 9.5 is a bound against the "constrained product" P_g^{A_k}
and Q_g^{A_k}. But since Phat_g and Qhat_g are built FROM those
constrained products (with a probability-kernel contraction from
Part 7.3 and B-marginal), the second moment bounds transfer by
kernel contraction of affinity (Section 9.1, property (c)).
LEMMA (one-group gain; cmp:one-group-gain). For each admissible
ordered pair (v, w) and each group index g,
H( Phat_g, Qhat_g ) >= 1 / (4 C_2).
WHAT IT SAYS. Every single-group contribution to the affinity is
at least a fixed positive constant 1/(4 C_2).
WHY IT MATTERS. Together with the tag normalization (Section 9.3)
and the counting (Section 9.7), this gives the modulus bound for
B(Phi).
PROOF. Combine:
(a) Lemma 9.1 (common finite measure): for two finite measures
P, Q with a witness nu of mass m > 0 satisfying the two
second-moment bounds C_P and C_Q,
H(P, Q) >= m^2 / sqrt(C_P * C_Q).
(b) From Section 9.5: nu-hat has mass m >= 1/2 against both
Phat_g and Qhat_g, with second-moment bounds C_P = C_Q =
C_2.
Hence:
H(Phat_g, Qhat_g) >= (1/2)^2 / sqrt(C_2 * C_2)
= 1 / (4 C_2).
This is uniform over g, (v, w), and the alpha window.
-----------------
9.7 COUNTING: THE MODULUS BOUND
-------------------------------
Now assemble.
We have shown, for every admissible ordered pair (v, w) and every
group g:
H(P_vw, Q_vw) >= (1/D) sum_g H(Phat_g, Qhat_g) [Section 9.3]
>= (1/D) sum_g 1/(4 C_2) [Section 9.6]
= G_D / (4 D C_2).
Recall from Part 7 that
G_D = floor( ell D / (4 M ) ),
so
G_D / (4 D C_2) >= ell / (32 M C_2).
So each ADMISSIBLE cell pair (v, w) contributes this lower bound
to the total occupation via the cell-affinity formula:
B(Phi) = (1/V^2) sum_{v, w} H( P_vw, Q_vw )
>= (1/V^2) * (number of admissible (v, w) pairs)
* ( ell / (32 M C_2) ).
From Section 8.8 (counting admissible pairs):
number of admissible (v, w) pairs >= upsilon_0 * V^2
= upsilon_0 * K^6,
with upsilon_0 = 1/2000.
Putting it together:
B(Phi) >= upsilon_0 * V^2 * (1/V^2) * ell / (32 M C_2)
= upsilon_0 * ell / (32 M C_2).
That is the modulus bound.
-----------------
PROPOSITION (uniform modulus occupation; cmp:modulus). After the
fixed choices of constants made above, there are D_0 < infinity
and
c_* := upsilon_0 ell / (32 M C_2)
= upsilon_0 ell / (32 M G C_path) > 0,
such that whenever D >= D_0, alpha = D r lies in
[alpha_0/2, 2 alpha_0], and K is sufficiently large, every
normalized nonnegative bosonic ground function satisfies
B(Phi) >= c_*.
The constant c_* is unchanged if the lower threshold on D is
subsequently increased. The volume thresholds can be taken
locally uniform for D in compact subintervals of [D_0, infinity).
WHAT IT SAYS. For any ground state (even a very complicated one),
the constant-orbital occupation is at least c_*, a fixed positive
number.
WHY IT MATTERS. This is the "modulus" half of the theorem. The
other half (Part 10 below) recovers the phase info to allow for
|Psi| to replace Psi.
PROOF. Summing the cell affinities and counting admissible pairs
as above. Uniformity is inherited from the uniformity of Parts
6-9, none of whose constants depends on D, K, or the ground
state.
-----------------
REMARK (what could go wrong, and does not). The proof works
because:
- The affinity cells are DISJOINT (each label contributes to
at most one cell affinity).
- The tag normalization has a fixed factor 1/D per group.
- The counting gives upsilon_0 of cell pairs admissible.
- The number of groups G_D per cell is at least ell D / (4M).
None of these uses independence between cells, or any
translation-invariance of the ground state, or any structure of
the ground vector beyond the identities used in Parts 5-6.
-----------------
9.8 WHAT COMES NEXT
-------------------
Part 10 does the "phase recovery" step (Stage E of the roadmap in
Part 2). We will:
- Construct, for each bath Y, a single-component subset U_Y of
the allowed slice S_Y that covers at least (1 - delta_geom) of
the slice's mass, with delta_geom small.
- Then apply the phase-transfer lemma (Part 3.5) with
A_Y = S_Y \ U_Y to obtain B(Psi) >= B(Phi) - 4 sqrt(delta_geom)
>= c_* - 4 sqrt(delta_geom).
- Choose D large enough that delta_geom is so small that the
bound is at least c_*/2.
The construction of U_Y is geometric: we will fill SMALL pockets
in each slice (components of S_Y of small diameter) and show that
the rest of the slice fits into at most one component (so its
mass defines U_Y).
--------------------------------------------------------------
PART 10 -- RECOVERING THE PHASES (THE GEOMETRY OF POCKETS)
--------------------------------------------------------------
10.1 WHAT GEOMETRY MUST PROVIDE
-------------------------------
Recall the phase-transfer lemma (Part 3.5): for each bath Y we
need a single-component subset U_Y of the allowed slice S_Y that
covers most of the slice's mass.
Specifically, we need:
delta_geom := int m(Y) * |A_Y| / V dY is small.
Here A_Y = S_Y \ U_Y is the discarded part, m(Y) is the bath
marginal (Part 2), and |.| is Euclidean volume.
The whole point: delta_geom -> 0 as D -> infinity (with alpha in
its fixed window). Then the phase-transfer bound
B(Psi) >= B(Phi) - 4 sqrt(delta_geom)
combined with the modulus bound (Part 9) is enough to conclude.
Part 10 constructs U_Y geometrically. The idea:
- Each ALLOWED SLICE S_Y can be complicated - its connected
components can be spirals, tubes, sponges.
- But the slice's components are constrained by Y: each
component is one of the connected regions of Lambda_K
minus the exclusion balls.
- If we "fill small pockets" of S_Y - components of SMALL
diameter - we get a simplified slice whose remaining
components are LARGE.
- If a component is large, we will see (via certificates)
that it is a "very rare" configuration.
So: PROVE the discarded volume is small.
10.2 THE LATTICE OF UNIT CUBES
------------------------------
A useful model: replace Lambda_K by a cubic lattice and track
which unit cubes are "free" (no particle at their boundary) and
which are "occupied" (near a particle).
Fix the grid of unit cubes B_v for v in L_K = (Z/KZ)^3 (Part 2,
Part 4.6). A cube is:
- BAD if the configuration puts particles in a pattern that
overlaps it (i.e., the certificate of the cell fails);
- GOOD otherwise.
Cell v is bad if it has no certificate (Part 4.6). Recall: the
joint probability that a prescribed set A of cells is bad is at
most p^{|A|} under the supply estimate (Part 7.5).
10.3 THE HOMEOMORPHIC GRID MODEL (geom:grid)
--------------------------------------------
To use the cubes for connectedness arguments, work at the level
of a lattice of cubes that models the SLICE topology.
Define, for a fixed bath Y, the BATH-SET
Q_cube(Y) = { v in L_K : B_v lies strictly inside the
union of radius-r closed balls
about the bath particles of Y }.
In other words, cube v is FULL of bath. And the FREE-SET:
Free(Y) = { v in L_K : B_v is disjoint from every radius-r
closed ball of Y }.
A cube may be FULL, FREE, or NEITHER (touched by a bath particle).
LEMMA (homeomorphic grid model; geom:grid). The slice S_Y, as a
topological space, is homeomorphic to a quotient of the free-set
Free(Y) by its adjacency. In particular:
- the connected components of S_Y correspond bijectively to
the max-connected components of Free(Y) (under nearest-
neighbor adjacency in the lattice);
- a component of S_Y has diameter at least a fixed positive
multiple of its corresponding cube-component's diameter
(both measured in appropriate units).
WHY IT MATTERS. Connectedness questions about slices become
questions about LATTICE connectedness, which we can control with
the combinatorial tools from Part 8.
The lemma is standard: the bath balls are disjoint from the
free cubes by definition, so any continuous path in S_Y stays
in the free part. Conversely, a lattice path of free cubes can
be made continuous, avoiding the baths, because nearest-neighbor
free cubes are adjacent along shared faces away from baths.
Boundaries are handled by choosing an appropriate refinement
(mesh finer than the exclusion distance).
10.4 FILLS AND EXTERIOR BOUNDARIES
----------------------------------
The essential geometric construction. Let C be a nonempty,
finite, max-connected set of cubes (e.g., one component of the
BAD CUBES).
FILL(C) := C union (every finite max-connected component
of Z^3 \ C).
So Fill(C) is C together with every FINITE hole it encloses.
Intuitively, we "fill in" the pockets surrounded by C.
The EXTERIOR BOUNDARY of C:
d_ext(C) := { z in Z^3 \ Fill(C) : z is adjacent (max-norm
distance 1) to some point of Fill(C) }.
This is the "outer shell" of C - the closest cubes you can stand
on without entering C or its finite pockets.
LEMMA (boundary lemma; geom:boundary). For nonempty finite
max-connected C:
(a) Fill(C) lies in the bounding box of C, expanded by 1 in
each coordinate.
(b) If C is not all of Z^3, then d_ext(C) is nearest-neighbor
connected.
(c) Every cube of d_ext(C) is disjoint from every closed
bad microcomponent (i.e., from all of C).
WHAT IT SAYS. The outer shell of C is a single connected
"slab" around C, disjoint from C itself.
WHY IT MATTERS. This is our tool for going AROUND bad clusters.
Given a bad set C in the way of a desired route, d_ext(C) lets
us reroute through safe cubes.
PROOF IDEA. (a) is by induction along tree paths in C. (b) uses
a lattice argument: for any two boundary cubes, connect them
through the exterior of Fill(C). (c) is by definition of d_ext.
10.5 SMALL FILLS: BOUNDING DISCARDED VOLUME
-------------------------------------------
We separate bad components into two classes.
DEFINITION. A finite microcomponent C of bad cubes is SMALL if
h * diam_inf(C) <= 1,
where h = 1/K is the microcube side and diam_inf is the max-norm
diameter in microcubes. So "small" means: its physical extent is
at most one unit cell.
LEMMA (small-fill volume; geom:small-fills). For K > 10 and
r <= 1, the projected union of the fills of SMALL bad components
has volume at most
C * r^2 * N,
with C an absolute constant.
WHAT IT SAYS. Physically, small bad components occupy only a
small fraction of the torus, of order r^2 N. Since D r^3 -> 0
(dilute regime), this is o(1) - a small volume.
WHY IT MATTERS. When we "delete" small fills from each slice,
the mass we lose is at most C r^2 N / V. In the dilute regime
(with r = alpha / D and V = N / D), this works out to a
vanishing quantity.
PROOF IDEA. There are two ingredients.
(i) A USEFUL COLUMN PROPERTY: for a finite bad component C, its
fill is contained in the bounding box of C by Lemma 10.4(a).
Moreover, every "column" parallel to coordinate 1 that meets
Fill(C) must also meet C. Otherwise the entire column would
avoid C and connect its own two ends to the two "sides" of C -
contradicting that C is finite-connected but encloses a finite
pocket.
WHY: if a column of cubes missed C entirely but intersected
Fill(C), then following the column upward, it would reach the
"top" of the fill without ever entering C. That contradicts
that Fill(C) is the union of C plus finite enclosed pockets.
So we can charge each fill-containing column to a distinct
point of C.
(ii) COLUMN VOLUME BOUND. Any such column meets the fill in at
most length (1 + 2h) by the bounding box. And each column has
cross-sectional area h^2 (one microcube). So, for each SMALL
component C:
|Fill(C)| <= h^2 * (1 + 2h) * |C| <= 3 h^2 |C|.
Summing over all small components:
|union of small fills| <= 3 h^2 * (total number of bad
microcubes in small components).
A closed radius-r ball meets at most C_0 = 343 = 7^3 microcubes
(because r/h <= 2 in the dilute regime). Since there are N-1
bath balls, the total bad microcube count is at most C_0 (N-1).
Using h = 1/K, h^2 = 1/K^2:
|union of small fills| <= 3 C_0 (N - 1) / K^2.
Since r = alpha / D with alpha bounded below by alpha_0 / 2,
and D r^3 = (alpha^3) / D^2 stays bounded, we have rK >=
const > 0 for every sufficiently large D in the regime. Absorbing
the constant into C:
|union of small fills| <= C' r^2 N.
Divide by V = K^3 = N / D:
|union of small fills| / V <= C' r^2 D = C' alpha^2 / D,
which tends to 0 as D -> infinity, uniformly in alpha in its
fixed window.
So SMALL components contribute a vanishing fraction of volume.
But we must also account for LARGE components. In the next
section we will see that large bad components force the
existence of many close pairs, which is a rare event. Large
components contribute a term that vanishes by Part 5.6's joint
spatial bounds.
10.6 LARGE COMPONENTS FORCE CLOSE PAIRS
--------------------------------------
A bad component is LARGE if its physical diameter is greater
than 1, i.e., if
h * diam_inf(C) > 1.
LEMMA (distinct witnesses in a large component; geom:large-witnesses).
There are absolute constants r_1, c_g > 0 such that the following
holds for r <= r_1 and K > 10. If a unit cell B_v contains a
microcube from a LARGE bad component, then at least c_g / r
DISTINCT bath particles lie within distance 5 of v, and each
has a distinct bath particle within distance 6r.
WHAT IT SAYS. A large bad component forces a LOCAL density of
particles: many particles within distance 5 of v, and each of
them has a very close partner (within 6r). So a large component
implies a violation of the "close count" condition of the
certificate.
WHY IT MATTERS. Close-count defects have small probability
(Part 5.6). Therefore large components are rare, and their
contribution to delta_geom is bounded by the p_D-type bound.
PROOF IDEA. Fix a lifted microcube z in the component, in a lift
of B_v. Walk through the component from z, stopping the first
time the scaled max-norm distance from z exceeds 1/4 (or 1, for
a "large" component). The walk contains at least 1/(4h) distinct
vertices and stays within distance 1/4 + h <= 1/2 of z.
Each such vertex yields a distinct bath microcube. Their physical
positions (the centers of these microcubes, scaled by h) give
distinct bath particles.
To see that each has a close partner within 6r: two microcubes
in the same *-connected bad component have lifts within max-norm
distance 1/4 of each other, hence physical distance at most 1/2
in the unit system (using r/h <= 2).
The key observation: each path vertex of the *-walk has a
NEARBY bath particle (the microcube contains, or is adjacent to,
one). Assign to each of the OTHER changed microcubes in the path
the particle closest to it. Two path vertices can share a
particle only when they are within max-norm distance c_g r of
each other. So at least 1/(4h) - O(1) >= 1/(8r) DISTINCT
witnesses survive, all within max-norm distance 5 of v, and each
with its own partner within distance 6r.
Set c_g = 1/8.
10.7 OPEN-SLICE CONNECTIVITY
----------------------------
The last geometric statement: after deleting small fills and
cells in the coarse bad set Q(X), the remaining part of each
slice is inside a SINGLE component.
Setting. Let (x, Y) be an admissible lift with no collisions
(guaranteed by r small and lift injectivity). Let
A(Y) = set of admissible positions x = S_Y,
with the "coarse bad set" Q(X) removed:
A(Y) := S_Y \ (union of the closed forbidden balls whose
parent cell lies in Q(X)).
(Here Q(X) is the coarse bad set of Part 5.6.)
LEMMA (open slice connectivity; geom:open-slice). There is a
uniform lower threshold D_0 such that, for D >= D_0, whenever
A(Y) is nonempty, it lies inside ONE connected component of S_Y.
Moreover A(Y) has the "strict clearance" property: every
continuous path found lies at strictly positive distance from the
closed forbidden balls.
WHAT IT SAYS. After removing cells in Q(X), the remaining part of
each slice is one connected region.
WHY IT MATTERS. This is the geometric input to the phase
transfer lemma. It lets us take U_Y = A(Y).
PROOF IDEA. Consider the LATTICE of complete-closed-unit-cubes
U_z that meet B_z and disjoint from all fill. Define the
exterior class
V = { z : U_z is disjoint from every fill }.
Claim: V is *-connected.
To see this, join successive cube centers through their common
faces. Each face piece is disjoint from every fill because the
cubes on both sides of it are in V and the fills avoid them.
Also, any continuous path drawn from the original point to its
cube center lies entirely in V: since the cube U_z is disjoint
from every fill, no opened fill can intersect a path staying
inside U_z.
A point on a grid FACE can be assigned to any adjacent cube:
membership outside the coarse A(Y) ensures that each such
cube is good and outside every fill.
Every closed good cube is disjoint from every closed forbidden
ball by the definition of "good". So the finite union of cubes
used by the route is compact, hence has a strictly positive
distance from the closed forbidden balls. All the line segments
between cube centers stay inside these cubes.
So A(Y) has both connectivity INSIDE ONE component and strict
clearance from the forbidden balls: A(Y) = S_Y minus fill is
connected, and A(Y) is at positive distance from the forbidden
balls.
The "close-count" and "distinct witness" lemmas (10.6) show the
removed volume is rare: it is contained in the union of small
fills (bounded by C r^2 N, Lemma 10.5) and the child cells of
Q(X) (bounded by p_D^{|A|}, Part 5.6).
-----------------
10.8 SMALL OMITTED VOLUME
-------------------------
Now assemble: delta_geom is small.
PROPOSITION (small omitted volume; geom:omitted-volume). Fix the
constants as above and take D sufficiently large. There are
absolute constants C, C_an such that, for K >= 100 and
C_an p_D <= 1/2:
delta_geom := int |A(Y)| / V nu(dY)
<= C r^2 D + C p_D + K^3 (C_an p_D)^{floor(K/10)}.
The sum tends to 0 as D -> infinity. The retained points belong
to ONE component of the open slice S_Y whenever the retained set
is nonempty. The estimate is uniform over the normalized ground
vector and alpha in [alpha_0/2, 2 alpha_0].
WHAT IT SAYS. The total volume we discard (delta_geom) tends to 0
as D -> infinity. So the phase-transfer argument recovers
B(Psi) from B(Phi) with a vanishing error.
WHY IT MATTERS. This is the final ingredient of the proof. With
delta_geom small we can make B(Psi) >= c_* - 4 sqrt(delta_geom)
arbitrarily close to c_*.
PROOF. Write delta_geom as a full-configuration expectation:
delta_geom = int_{Lambda_K^N} |A(x, Y)| / V mu(dx dY).
Here mu is the full N-particle measure. A(x, Y) is the "retained
set" for a bath Y with the removed ball of x.
Now stratify the integrand according to what has been removed:
- SMALL fills (diam <= 1): volume <= C r^2 N by Lemma 10.5,
and this bound is DETERMINISTIC. Contribution: C r^2 N/V =
C r^2 D.
- "Coarse bad" cells (Q(X)): the set Q(X) has probability
p_D^{|A|} jointly (Section 5.6). Contribution: at most the
probability that x's own cell is in Q(X), which is at most
C p_D.
- LARGE components: have a *-component of size >= m_K =
floor(K/10). By the counting lemma (Section 8.6),
the probability that the bad set Q(X) has a *-component of
size >= m_K is at most
K^3 (C_an p_D)^{floor(K/10)}
by the counting lemma (Section 8.6). Contribution: the same
quantity.
Adding the three terms:
delta_geom <= C r^2 D + C p_D + K^3 (C_an p_D)^{floor(K/10)}.
Why each term tends to 0 as D -> infinity (with alpha in its
fixed window):
- C r^2 D = C alpha^2 / D -> 0 (alpha fixed, D grows).
- C p_D = C (C_theta b^{-2} D^{-4/3})^{1/Q} -> 0
(p_D -> 0 by Section 5.6).
- K^3 (C_an p_D)^{floor(K/10)} -> 0: K^3 is polynomial in K
while (C_an p_D)^{floor(K/10)} decays like a fixed small
base raised to a K-proportional power.
In particular, the retained set A(Y) = S_Y minus the removed
region lies in ONE connected component of S_Y (Lemma 10.7),
and its discarded volume fraction is delta_geom, which we have
just shown is vanishing.
CONCLUSION OF PART 10. The geometric input for the phase-transfer
lemma is available: for sufficiently large D (with alpha in its
fixed window, K large), delta_geom is smaller than any prescribed
positive number. In particular, it can be made smaller than
(c_* / 8)^2, so that the correction term 4 sqrt(delta_geom) in
the phase-transfer lemma is at most c_* / 2.
--------------------------------------------------------------
PART 11 -- PUTTING IT TOGETHER AND TAKING THE LIMIT
--------------------------------------------------------------
11.1 ORDER OF THE CONSTANTS
---------------------------
One of the less-glamorous but most important steps: pick the
constants in the RIGHT ORDER so that every earlier estimate holds
simultaneously.
The reading order in Parts 5-10 is not the pick order. Here is
the pick order. It matters because a later choice may need to
be large enough to accommodate an earlier-bound error, but must
not invalidate an estimate that was made before.
(P1) FIX alpha_0 large enough for the density-uniform bounds of
Part 5 (low-count and close-count defects).
(P2) FIX ell small (Section 5.4) to match the threshold on
particle counts per cell.
(P3) FIX theta, then b small (Section 5.6) for the joint
close-count bounds.
(P4) FIX t > 0 small (Part 6) for the bridge estimates - the
exponent exp(-c/t) is arbitrarily close to 1 for t small.
(P5) FIX the walk length parameter R = 30 and pick the local
comparison bound G >= 2 (Part 7.6).
(P6) FIX the integer group size M (Part 7) large enough for
the grouping failure estimate: 2(G-1)/M <= kappa/2.
(P7) FIX D_0 large enough for all the preceding estimates AND
for ell D_0 >= 8M and -4 log(1 - chi_D) <= kappa/2.
Then we can define
c_* = upsilon_0 ell / (32 M C_2)
with C_2 = G C_path. None of upsilon_0, ell, M, G, C_path
changes when the lower threshold on D is subsequently increased
- only D grows.
11.2 THE MODULUS IN THE SCALED VARIABLES
----------------------------------------
STAGE E of the roadmap (Part 2): recover the phase loss of Stage
A. We have shown (Part 9) B(Phi) >= c_* for nonnegative ground
states Phi = |Psi|. And we have shown (Part 10) that the
discarded volume delta_geom is small - eventually smaller than
(c_*/8)^2.
Now apply the phase-transfer lemma (Part 3.5):
B(Psi) >= B(Phi) - 4 sqrt(delta_geom)
>= c_* - 4 sqrt(delta_geom).
>= c_* - 4 (c_* / 8)
= c_* - c_* / 2
= c_* / 2.
So:
PROPOSITION (main:scaled). There are absolute constants
alpha_0, D_*, c_* > 0 such that the following holds. For
D = N/K^3 >= D_*, with r = alpha / D and alpha in
[alpha_0/2, 2 alpha_0], for all sufficiently large integer K,
EVERY normalized complex bosonic ground vector satisfies
B(Psi) >= c_* / 2.
The volume threshold D_* can be chosen uniformly when D ranges
over a fixed compact subinterval of [D_*, infinity), and is
independent of the chosen ground vector.
-----------------
COROLLARY (corollary in the original). Under the hypotheses of
the proposition, the same bound holds for every density operator
supported on the bosonic ground space; in particular for the
normalized ground-space projection, and for every sequence of
nonnegative ground functions.
This is immediate: a density operator on the ground space is a
weighted average of projections |Psi_i><Psi_i|, and B is linear
in the density matrix.
11.3 THE THERMODYNAMIC LIMIT
-----------------------------
Now translate back to the original problem.
We are given a sequence (N_k, L_k, Psi_k) with N_k / L_k^3 -> rho,
ground vectors Psi_k, and a fixed scattering length a > 0. The
dilution hypothesis is rho a^3 < eps_0 for a small fixed eps_0.
Set alpha_0 and D_* as above. Define, for all sufficiently large
k:
q_k = sqrt(a N_k / (alpha_0 L_k)),
K_k = floor(q_k),
r_k = a K_k / L_k,
D_k = N_k / K_k^3.
The point of this choice is to keep the "scaled density" alpha
close to alpha_0.
Compute:
alpha_k := D_k r_k
= (N_k / K_k^3) * (a K_k / L_k)
= a N_k / (L_k K_k^2).
Using K_k^2 = floor(q_k)^2 = q_k^2 (1 + O(1/q_k)):
= a N_k / (L_k q_k^2) * (1 + O(1/q_k))
= a N_k / (L_k * a N_k / (alpha_0 L_k)) * (1 + O(1/q_k))
= alpha_0 * (1 + O(1/q_k))
so alpha_k -> alpha_0. In particular alpha_k lies in
[alpha_0/2, 2 alpha_0] for large k.
For D_k, use K_k = q_k (1 + O(1/q_k)) and q_k ~ sqrt(a rho L_k^3 / (alpha_0 L_k)) = L_k sqrt(a rho / alpha_0):
K_k ~ L_k sqrt(a rho / alpha_0).
Then
D_k ~ N_k / (L_k sqrt(a rho / alpha_0))^3
= rho L_k^3 / (L_k^3 * (a rho / alpha_0)^{3/2})
= (alpha_0^{3/2}) / sqrt(rho a^3).
As L_k -> infinity with N_k / L_k^3 = rho, this converges to
D_infty := alpha_0^{3/2} / sqrt(rho a^3).
Dilution (rho a^3 < eps_0) and alpha_0 large together imply
D_infty > 2 D_*. Absorbing D_* into eps_0, we have D_k >= D_*
for all large k.
-----------------
ASSEMBLING THE MAIN THEOREM. For all large k, the pair
(D_k, alpha_k) is in the regime of Proposition main:scaled. So
B(Psi_k) >= c_* / 2 for all sufficiently large k.
Take liminf:
liminf_{k -> infinity} B(Psi_k) >= c_* / 2 =: c_0.
This is exactly the statement of the main theorem, since
liminf >= c_0 > 0 means a fixed positive fraction of particles is
in the constant orbital, in the limit.
COROLLARY (mixed states). Every density operator supported on the
bosonic ground space has the same lower bound. Rigid phase-locking
and microscopic inhomogeneity are both excluded.
CONCLUSION. We have proved: dilute hard-sphere Bose gases in the
thermodynamic limit necessarily have a fixed positive fraction of
their ground-state particles in the constant orbital, uniformly
over complex phases of the ground vector, over boundary-condition
choices, and over the exact density. The lower bound is not
quantitative (no explicit numerical value of c_0), but it is
uniform in the volume.
--------------------------------------------------------------
APPENDIX A -- GLOSSARY
--------------------------------------------------------------
Every term the main text introduces, defined concisely. Sorted
alphabetically.
AFFINITY H(P, Q)
For two finite nonnegative measures P, Q on the same space,
H(P,Q) = integral of sqrt(p q) dxi, where p, q are the
densities relative to any common dominating measure xi. H is
invariant under choosing another xi. Interpretations:
- the "geometric mean" of the two measures' mass,
- a measure of statistical overlap between two
distributions.
Used throughout Part 9 as the overlap between "before" and
"after" cell-pair measures. Range: 0 <= H <= sqrt(||P|| * ||Q||).
ALLOWED SLICE S_Y (or S_Y)
For a fixed bath Y = (x_2, ..., x_N), the open subset of
Lambda_K consisting of positions x that can be added without
violating the exclusion (i.e., x is at distance > r from
every bath particle). See Section 3.4.
BAD CELL / BAD SET
A cell v (unit cube B_v) is BAD if it lacks a certificate
(Part 4.6). A lattice point z in the microcube grid is BAD
if the corresponding microcube is within distance r of a bath
particle. The BAD-SET ASSUMPTION in Part 8 states that for a
random bad set B and any A subset L_K,
P(A subset B) <= p^{|A|}. This is exactly the joint supply
property that Part 5.6 provides.
BEC (BOSE-EINSTEIN CONDENSATION)
The phenomenon in which, at sufficiently low temperature, a
macroscopic fraction of a bosonic gas occupies the single
lowest-energy one-particle state. See Part 1.
BRIDGE (BROWNIAN)
A Brownian path conditioned to start at position y at time 0
and end at position u at time 2t. The law is Gaussian; see
Sections 4.1 and 6.1. Every "trial" in Part 6 is a bridge
through a chosen midpoint u.
CELL B_v
The unit cube v + [-1/2, 1/2]^3 in the torus Lambda_K, indexed
by v in L_K = (Z/KZ)^3. Its volume is 1 and the cells tile
Lambda_K. See Section 2.4.
CERTIFICATE AT v
Six conditions (see Section 4.6) that make a cell usable for
the local rerouting argument:
- n_v(Y) >= ell D (many particles in cell v);
- C_v(Y), C_v(Z) < theta D (few close-parents at both ends);
- few visiting trajectories (at most H_1 D);
- each visitor is regular;
- few visitors with lifted diameter > 1;
- the same conditions with Z in place of Y.
A cell with no certificate is called bad.
CLOSE-COUNT
For a configuration X and a site v,
C_v(X) = #{ i : d(X_i, v) <= 5 and d_i(X) <= s_0 },
where d_i(X) is the distance from X_i to its closest other
particle. So C_v counts particles near v that also have a
close partner. Bounded-close-count is one of the certificate
conditions. Its joint probability is bounded by p_D^{|A|}
(Section 5.6).
CONSTANT ORBITAL varphi_0
The uniform one-particle wave function varphi_0(x) = 1/L^{3/2}
(or 1/V^{1/2} in scaled units). The one-particle state that
is spread evenly over the torus. "BEC" in the sense of this
paper means "many particles in varphi_0".
D (SCALED DENSITY)
D = N / V, the average number of particles per unit cell in
the scaled torus Lambda_K. In the physical system, D is only
a bookkeeping device; in the thermodynamic limit D -> D_infty
= alpha_0^{3/2} / sqrt(rho a^3).
delta_geom
The discarded volume fraction from the phase-recovery step
(Part 3.5, Part 10). Defined as
delta_geom = integral of m(Y) * |A_Y| / V dY (nu)
where A_Y = S_Y minus the retained subset U_Y. The whole
point of Part 10 is delta_geom -> 0 as D -> infinity.
ELIGIBLE LABEL
A label i whose trajectory omega_i is "movable" at cell v
(Section 7.1):
- initial endpoint Y_i in B_v;
- final endpoint |Z_i - v| <= 3;
- initial and final endpoints are s_0-separated from all
other labels;
- trajectory is regular (Part 4.4) and confined to the
radius-10 ball around v.
EXCLUSION DISTANCE r
The hard-wall minimum distance between any two particles.
Physically it is the scattering length a; in scaled units
it is a / (cell side), and equals r = alpha / D. Particles
may not come closer than r.
FORM q_N
The Dirichlet form q_N[F] = integral of |grad F|^2 over the
N-particle allowed configuration set, extended by zero to
the boundary. This is the quadratic form for the hard-sphere
Hamiltonian (see Section 2.1).
GAS PARAMETER
The dimensionless combination rho a^3, where rho is
density and a is scattering length. In these units
rho a^3 = D r^3 = alpha^3 / D^2. The dilute regime is
rho a^3 -> 0, or equivalently D -> infinity at fixed alpha.
GROUND STATE
A normalized wave function Psi minimizing the form q_N
restricted to the bosonic (symmetric) subspace. Its energy is
E_N. See Section 2.2.
IMS
The "IMS localization formula" (Section 5.5): integrate a
probability-weighted partition of unity into the form,
gaining an extra term from the gradient of the partition.
Named after Ismagilov, Morgan, and Simon.
K
Integer side length of the scaled torus Lambda_K = (R/KZ)^3.
Volume V = K^3. Chosen at the end so that D and alpha land in
the right window. Related to N and D by V = N/D.
LABEL
An abstract index i in {1, ..., N}, tracking the "identity"
of a particle through time. All particles are identical
bosons, but the combinatorics of labels is what makes the
path-integral construction tractable.
LAMBDA_K, LAMBDA_L
Lambda_K = (R/KZ)^3 is the scaled flat torus that the proof
actually works on. Lambda_L = (R/LZ)^3 is the original
physical torus.
M (GROUP SIZE)
The integer number of labels per group in Part 7. Fixed once
and for all after G and before D. Appears in the definition
G_D = floor(ell D / (4M)) of the number of groups per cell.
mu
The probability measure on N-particle configurations defined
by mu(dX) = Phi(X)^2 dX, where Phi = |Psi| and Psi is a
normalized ground state. Every probabilistic bound in Parts
5-10 is a bound under mu.
nu-hat (COMMON MEASURE)
The finite measure on (B, x, y, Y) constructed in Section 9.4
as a "shared witness" between the two midpoint pictures. Its
total mass is at least 1/2 (Lemma 9.4). Domination of nu-hat
by both Phat_g and Qhat_g is the key to the affinity bound.
OCCUPATION FRACTION B(Psi)
For a normalized many-particle wave function Psi, the number
B(Psi) = (1/V) * integral over (x, y, Y) of
Re[ Psi(x, Y) conj(Psi(y, Y)) ] dx dy dY
(equals (1/V) * integral of |int Psi dx|^2 dY when Psi is
nonnegative). It is the fraction of particles in the constant
orbital, in [0, 1]. See Section 2.3.
ONE-PARTICLE DENSITY MATRIX Gamma^{(1)}
Gamma^{(1)}_Psi(x; y) = N * integral over Y of
Psi(x, Y) conj(Psi(y, Y)) dY. Positive semi-definite, trace N.
The occupation B(Psi) is <varphi_0, Gamma^{(1)} varphi_0> / N.
PATH MEASURE R (STATIONARY KILLED PATHS)
A probability measure on continuous paths omega = (omega_1,
..., omega_N) : [0, 2t] -> Lambda_K. At each time its marginal
is mu (Sections 4.1-4.2). Provides the temporal structure that
lets us "replace" a piece of a single particle's trajectory.
P_i (REFERENCE LAW)
The single-path reference measure for label i:
P_i(d omega_i) = 1_{C_i(F)} (d B_i / a_i),
where B_i is a free bridge of duration 2t and C_i(F) imposes
regularity, confinement, and avoidance of the frozen paths F.
See Section 7.3.
Pi_i^U (PROPOSAL LAW)
The single-path proposal: choose midpoint u uniformly in a
subset U of a target cell, sample a two-leg bridge through
u, and condition on success against frozen paths. See
Section 7.6.
Psi, Phi
Psi is a (possibly complex) normalized ground vector. Phi =
|Psi| is its modulus, also a ground vector (Lemma 3.2).
Q(X) (COARSE BAD SET)
The set of cells v with C_v(X) >= theta D (too many close
pairs nearby). Its joint probability is at most p_D^{|A|}
(Proposition 5.6). Q(X) is rare and forces the discarded
volume to be small.
REGULAR PATH
A path satisfying three bounds (Section 4.4):
- mesh-by-mesh diameter bounded by D^gamma;
- average (1+A_j)^3 bounded by C_reg;
- outer windows have lifted diameter <= s_0/8.
Failure of regularity has probability at most C_q D^{-q}
for any q under free Brownian motion.
rho
Physical density N/L^3. Related to the scaled density by
D r^3 = rho a^3.
s_0 (ENDPOINT SEPARATION SCALE)
s_0 = b D^{-1/3}. The scale below which two endpoints are
considered "the same" and at which labels are declared
s_0-separated. Chosen so that the average number of
s_0-balls per cell is small (b fixed small).
SCATTERING LENGTH
See EXCLUSION DISTANCE. Physically, the length a such that the
two-body zero-energy scattering solution is 1 - a/|x|.
SLICE S_Y
See ALLOWED SLICE S_Y.
upsilon_0
The constant 1/2000. The fraction of ALL ordered cell pairs
that are "admissible" in the sense of Section 8.8. It appears
in c_*.
V
Volume K^3 of the scaled torus Lambda_K. V = N/D.
varphi_0
See CONSTANT ORBITAL.
Y, Z
Endpoint configurations of a path sample. Y = X(0), Z =
X(2t). In Part 7, (Y_i, Z_i) are the specific endpoints of
label i's trajectory.
--------------------------------------------------------------
APPENDIX B -- FURTHER READING
--------------------------------------------------------------
If you want to learn more about Bose-Einstein condensation, hard
spheres, or the specific techniques used here, these are good
starting points. All citations refer to the bibliography of the
original paper.
FOR THE PHYSICS OF BEC
- Penrose and Onsager 1956. Introduced the "off-diagonal
long-range order" definition of BEC - a precursor to the
density-matrix language used throughout this paper.
- Lee, Huang, and Yang 1957. Computed the first quantum
correction to the ground-state energy of a dilute hard-
sphere Bose gas. The "Lee-Huang-Yang" formula is the
conjecturally exact condensate depletion.
- Lieb and Seiringer 2002. Rigorous proof of 100% BEC (in
the density-matrix sense) for the dilute trapped Bose gas.
This paper's target is the same theorem in the translation-
invariant case, with a fixed fraction instead of 1.
- Bogoliubov 1947. The original "Bogoliubov approximation"
behind the modern theory of the dilute Bose gas.
FOR THE HARD-SPHERE MATHEMATICS
- Dyson 1957. The "Dyson lemma", which is the general
mechanism used here for inserting particles without
paying too much energy cost.
- Lieb and Yngvason 1998. Ground-state energy of the low
density Bose gas; the "Neumann subdivision" method that
underlies the local crowding bound (Part 5.1).
- Lundholm, Portmann, Solovej 2015. The two-particle
coercivity used in Part 5.2 for the close-neighbor
energy.
- Basti, Brooks, Cenatiempo, Olgiati, Schlein 2026.
Rigorous upper bound for the Lee-Huang-Yang energy of
hard spheres. The matching lower bound is the
condensate-fraction theorem proved here.
FOR THE PROBABILISTIC IDEAS
- Benjamini, Pemantle, Peres 1998. The "unpredictable
paths" paper, whose path-construction and encounter-
moment bounds are imported in Part 8.
- Peres and Sly 2014. Rigidity and tolerance for perturbed
lattices.
- Arias-Castro, Candes, Helgason, Zeitouni 2008. A
statistical view of traversing a lattice through a sparse
set of obstacles.
FOR THE MATHEMATICAL BACKGROUND
- Faris and Simon 1975. Degenerate and non-degenerate
ground states for Schrodinger operators - the source
of the positivity, modulus, and phase facts in Part 3.
- Frank and Seiringer 2008. The ground-state multiplier
identity used throughout Part 5.
- Fournais 2021. Length scales for BEC in the dilute Bose
gas - a short review that puts this paper in context.
- Solovej 2025. A survey of dilute Bose gases that can be
read as the state of the art before this paper.
FOR THIS PAPER'S SPECIFIC COMPANION
- The companion article "A density-uniform condensate bound
for dilute Bose gases" (OpenAI Math Release, September
27, 2026). The only imported results are the two moment
bounds of Section 8.4 (IMP1 and IMP2). Everything else is
proved in the present document.
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