A PIVOTAL AMPLITUDE FOR VORONOI PERCOLATION
FROM CARDY'S FORMULA
A self-contained proof, written to be read.
Conditional on the scalar Cardy hypothesis.
Main result: E N_eps ~ c_V * eps^(-3/4), c_V in (0, infinity).
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READING GUIDE
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The paper proves one theorem. Everything else is machinery, and each
section says up front which piece of the theorem it feeds. If you lose
the thread, come back here.
PART I The problem and the answer, and why eps^(-3/4) is expected.
PART II What we are allowed to use: the companion paper's results.
PART III Four-interface transmission: the base quantity that makes
the proof work, and the cap identity that computes with it.
PART IV Large-modulus asymptotics of that transmission. A sleep
of the four-interface operator, and the two frozen degrees
of freedom whose decay buys integrability.
PART V Separating the inner and outer conformal data of an annulus.
PART VI Making the two-end estimates uniform over arbitrary signed
boundary data, and iterating them in the discrete model.
PART VII Differentiating the insertion probability in the Poisson
intensity. Two insertions far apart, then two insertions
of vastly different aspect, plus the tail estimate. This
is where the amplitude comes from.
Every lemma is proved in the body. The only inputs we do not prove are
collected in Part II; they are quoted from the companion preprint [V].
Wherever the proof uses a result of [V], the reference is spelled out
in words and a locator is given.
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PART I. THE PROBLEM AND THE ANSWER
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1. What is being counted
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Place a random set of points in the plane according to a Poisson process
of intensity eps^(-2). Draw the Voronoi tessellation: the cell of each
nucleus is the set of plane points closer to that nucleus than to any
other. Color every cell independently black or white with probability
1/2 each. This is critical planar Voronoi percolation.
Let Q be a bounded quadrilateral: a topological disk with four
distinguished boundary points in cyclic order. A black left-right
crossing of Q means a connected cluster of black cells entirely inside
Q that joins the two opposite boundary arcs between the marked points.
Likewise for white. Since Q is not resampled at the boundary, we use
the full-plane tessellation and closed cells.
A nucleus is color-pivotal for a black crossing if recoloring its cell
(only its cell, leaving every other cell fixed) switches the event
"there is a black left-right crossing" from happening to not happening,
or from not happening to happening. Write
N_eps(Q) = number of color-pivotal nuclei.
This counts nuclei including those outside Q if their cells meet Q.
Since each pivotal cell has to reach in among the arms that decide the
crossing, N_eps(Q) is a local quantity.
For the square [0,1]^2 with a left-right crossing, write N_eps = N_eps(Q).
The random variable N_eps depends on both the point pattern and the colors.
Every expectation below averages over both.
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2. The main theorem
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The one hypothesis we do not prove is Cardy's formula for Voronoi percolation.
State it before the theorem so there is no ambiguity about what is assumed.
ASSUMPTION (scalar Cardy limit). For every bounded Jordan quadrilateral Q
with four distinct marked boundary points, the critical crossing probability
converges, as eps goes to zero, to Cardy's conformally invariant value.
Here "Jordan quadrilateral" means a Jordan domain with four marked points
in cyclic order and a specified pair of opposite boundary arcs.
In the upper half-plane with marked points 0, x, 1, infinity and 0 < x < 1,
the crossing between [0, x] and [1, infinity] has Cardy value
(Gamma(2/3) / (Gamma(1/3) Gamma(4/3))) * x^(1/3) * 2F1(1/3, 2/3; 4/3; x).
The Assumption is only about a single crossing event at a time. It does
NOT claim a rate of convergence. It does NOT claim an SLE or multi-crossing
scaling limit. Those would be extra hypotheses we avoid.
THEOREM. Under the scalar Cardy Assumption, there exists a constant
c_V in (0, infinity), depending only on the square, the crossing, the
intensity convention, and the color-pivotal counting rule, such that
lim eps^(3/4) E N_eps = c_V.
eps->0
Equivalently, the inverse expected pivotal count satisfies
(E N_eps)^(-1) ~ c_V^(-1) eps^(3/4).
We do not compute c_V. We prove that the amplitude exists and is finite
and positive.
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3. Why eps^(-3/4) is the right order
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The exponent is not new. It follows from the alternating four-arm exponent
of critical percolation, which is 5/4. Here is the logic.
The unit square [0,1]^2 has about eps^(-2) nuclei. For a nucleus to be
color-pivotal for a crossing of the square, the four interfaces emanating
from the boundary of its cell -- two black, two white, alternating around
its boundary -- must survive from the cell boundary all the way to the
scale of the square. This is called "four alternating arms." For critical
triangular-lattice percolation, Smirnov and Werner proved the alternating
four-arm probability A(r, R) between annuli of radii r < R is of order
(R/r)^(-5/4): the exponent is 5/4.
For a nucleus at a typical point of the square, the "inner radius" is the
size of its cell, which is of order eps (the typical Voronoi cell diameter
at intensity eps^(-2)). The "outer radius" is of order 1, the square's
size. So A(eps, 1) is of order eps^(5/4). Multiply by the eps^(-2) nuclei:
E N_eps is of order eps^(-2) * eps^(5/4) = eps^(-3/4).
That is the order. The theorem says the proportionality constant converges;
it is not a slowly varying or oscillating factor.
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4. The distinction between exponent and amplitude
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Having E N_eps = eps^(-3/4 + o(1)) still allows a slowly varying or
oscillating multiplicative correction. An amplitude theorem pins down
the correction: it says eps^(3/4) E N_eps converges to a specific finite
positive number. This is strictly stronger than the exponent and strictly
weaker than computing c_V.
For critical triangular-lattice percolation, sharp arm asymptotics -- with
a limiting multiplicative constant -- were proved by Du, Gao, Li, and
Zhuang. Our theorem is of the analogous type but for a Poisson-Voronoi
color-pivotal count, and it is conditional on the scalar Cardy hypothesis.
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5. The strategy of the proof
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The result is a statement about the size of a "typical" pivotal region.
The strategy is to turn it into a statement about how a pivotal count
responds to changes in the Poisson intensity, and then integrate the
response.
We work throughout at unit intensity and rescale at the end. Let
Q_0 = [-1, 1] x [-1/2, 1/2],
f(R) = P(a freshly inserted nucleus at the origin is
color-pivotal for the left-right crossing in R Q_0),
where R Q_0 is the R-scaled rectangle. The color is a fair coin; the
word "nucleus at the origin" means add one unit-intensity point.
Two different exponents describe f:
(a) from Part II, f(R) = R^(-5/4 + o(1));
(b) from Part VII, the logarithmic derivative of f is -5/4 plus an
integrable error.
Together (a) and (b) show f(R) = a * R^(-5/4) * (1 + o(1)) for some
constant a in (0, infinity). Calibration then converts this into the
amplitude for E N_eps.
The reason we bother with f rather than a four-arm probability is that
f has a second representation. Since f(R) = probability that the origin
is pivotal, we can differentiate f in the intensity of the Poisson
process and get a kernel K_R(y) that measures how a fair-colored nucleus
placed at position y changes f(R). Integrating K_R(y) over y recovers
the logarithmic derivative of f:
(R/2) d/dR f(R) = integral over plane of K_R(y) dy.
(This is equation (cl:derivative-eq) in Part VII, Lemma 6.1.)
To show this logarithmic derivative equals -5/4 plus integrable error,
we need two analytic controls on the transfer of signed connection
states across an annulus around the origin.
First control: FAR insertions. A nucleus inserted at radius R from the
crucial region is almost invisible if R is large. The message its
insertion carries to a cut at radius t << R has size O(t^(-2-nu)) for
some nu in (0, 1). This is Corollary tr:insertion in Part VI.
The exponent -2 - nu is the crucial first fact: it is strictly faster
than t^(-2), which makes the insertion response integrable over the
two-dimensional plane.
Second control: NEAR insertions at two outer scales. Two rectangles
RQ_0 and R'Q_0, with R' in [R, 2R], each pass through a common set
of cut circles around the origin. The R-transfer and R'-transfer
through these cuts share all intermediate messages, so their difference
is controlled by the rank-one character of the transfer. This is
Corollary tr:wedge in Part VI.
The rest of the paper builds those two controls. Part III constructs
the annular transmission as a conformal-limit object. Part IV controls
it at large modulus. Part V strips it down to depend separately on
inner and outer data. Part VI wraps it in a uniform operator estimate.
Part VII uses the operator estimates to compute the derivative.
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6. What each Part delivers to the previous Part
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Deliverable of Part III: the annular transmission function
T^pm(p, x, y),
p = log(outer/inner) modulus, x, y the four-switch data on each end,
defined as the probability that all four interfaces cross annulus
when the inner cap state is fixed to +/-. It is continuous in (p, x, y),
it extends by zero at switch collisions, it is invariant under simultaneous
rotation and global color reversal, and cyclic relabeling interchanges
+ and -.
Deliverable of Part IV: for large p,
T^pm(p, x, y) = e^(5p/4) (K h(x, y) + O(e^(-sigma p))),
where K in [0, infinity) is a universal positive constant, h(x, y) =
J(x)^(1/3) J(y)^(1/3) with J a Vandermonde-like product of sines, and
sigma > 0. This is the rank-one asymptotic in the annular uniformization.
Moreover, the SIGNED combination (T^+ - T^-), with its inner end
averaged over rotation, decays like e^(-9p/4), a power strictly faster
than e^(-2p).
Deliverable of Part V: after normalizing each end separately by its
logarithmic capacity and its marked angles, the SAME asymptotic formula
holds with the modulus p replaced by log(l_o/l_i) and the angles by the
separately normalized angles, up to errors of order (l_i/l_o)^(5/4 + sigma_1).
The signed rotation average decays like (l_i/l_o)^(2 + sigma_2).
Deliverable of Part VI: transfer operators M_{r,s} between seam data at
radii r < s. For a fixed annulus ratio L = s/r much larger than one but
of fixed size, the transfer M_{r, Lr} satisfies:
||M|| <= C L^(-5/4),
M = M_1 + E, rank(M_1) <= 1, ||E|| <= C L^(-5/4 - delta),
||M on even-isotropic inputs|| <= C L^(-2 - delta).
Here "even" means invariant under global color reversal, "isotropic"
means invariant under rotation, and delta > 0. Iterating these gives
Corollary tr:insertion (fair-insertion message, exponent -2 - nu)
and Corollary tr:wedge (contraction of antisymmetric products).
Deliverable of Part VII: the amplitude theorem.
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7. What is NOT new
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The passage from local crossing data to joint crossing information uses
Schramm-Smirnov's quad-crossing framework and gluing theory. The annular
exploration calculation uses Zhan's annulus SLE. Box-crossing and arm
estimates for Voronoi percolation are from Tassion and Vanneuville.
Sharp arm asymptotics for triangular percolation and the SLE6 /
Calogero-Sutherland connection are from Smirnov-Werner, Cardy, and DGLZ.
Weights for pivotal measures are from Garban-Pete-Schramm. The quenched
near-critical theory we invoke is the companion paper [V].
What is new here: (i) the explicit reduction of the color-pivotal
amplitude to (a) a four-interface transmission in an annulus, and (b)
an intensity derivative; (ii) the rank-one asymptotic and the
rotation-averaged signed estimate for T^pm; (iii) the uniform transfer
operator machinery on arbitrary signed connection states; (iv) the tail
estimate that makes the intensity derivative summable.
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PART II. KNOWN INPUTS
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Everything in this Part is quoted from the companion preprint [V].
Nothing here is proved in this paper. We collect the statements under
names we will use later. The rest of the paper relies only on these.
Throughout we work at UNIT Poisson intensity (Voronoi cells come from
a unit-rate Poisson process in the plane). The mesh parameter appears
only when we translate back to eps^(-2) intensity, via scaling.
Conventions. C, c > 0 denote constants whose values may change between
occurrences. u \asymp v means the ratio u/v lies in some fixed interval
[a, b] with 0 < a < b < infinity. Fixed-factor changes in arm radii and
fixed buffer widths are absorbed in C. Let b_* > 0 be a fixed
microscopic scale (think: much larger than the mesh but much smaller
than any macroscopic distance).
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8. Arms, screens, and local change
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An "arm" in an annulus is a monochromatic cell path joining the two
boundary components. "Four alternating arms" means four arms, colored
black, white, black, white in cyclic order around the annulus.
Let A(r, s) denote, for b_* <= r < s, the probability of four alternating
arms in the square-annulus convention of [V].
INPUT A (arms). There exist g in (1, 2), alpha > 0, beta > 0, and
constants C, c > 0 such that for b_* <= r < s < t:
(A1) c (r/s)^g <= A(r, s) <= C (r/s)^(1 + alpha).
(A2) C^(-1) A(r,s) A(s,t) <= A(r,t) <= C A(r,s) A(s,t).
(A3) P(one-arm from radius r to s) <= C (r/s)^beta.
(A4) P(three alternating arms in a straight half-plane
from radius r to s) <= C (r/s)^2.
In words: alternating arms decay with a power between two exponents, the
decay is quasimultiplicative across two annuli, one-arm and straight
boundary three-arm events decay quickly enough.
These are Lemmas 2.1 and 2.3 of [V]. The quantitative arm theory builds
on the box-crossing and annealed arm estimates for Voronoi percolation
(Tassion, Vanneuville).
INPUT B (screens / localization). Fix a collar of diameter comparable to
s (formal: the annulus between radii (1 - eta)s and (1 + eta)s for a
small fixed eta > 0). There exists an event G_s, determined by the Poisson
points in a SLIGHTLY LARGER collar, such that:
(B1) G_s is rotation-equivariant about the center of the annulus.
(B2) On G_s, the graph and every vertex star inside the smaller
collar are determined by those points, with mesh smaller than
any prescribed fixed fraction of s.
(B3) P(G_s^c) <= C exp(-c s^2).
(B4) Events G_s with disjoint enlarged supports use independent
Poisson variables.
In words: with exponentially high probability, everything inside the
smaller collar is "local" to Poisson points in the larger collar, and
those Poisson points are independent because the larger collars
are disjoint.
An event G_s with these properties is called a SCREEN. A concrete
example: require that every ball of a small fixed relative radius (say
eta/10)s inside the collar contain at least one nucleus. This is
rotation-equivariant and has exponential failure probability by a finite
covering and the Poisson empty-ball formula. Extra buffers between the
two collars ensure B2. A nucleus inserted beyond the enlarged support
cannot send its cell through the screened collar, hence cannot influence
the graph in the smaller collar.
We also allow FAILED screens inside an ARM PROBE; the probes below use
a convention due to [V]:
INPUT C (screened arm probes). Place screens at geometrically spaced
radii r, Lr, L^2 r, ... where L > 1 fixed. If the largest failed screen
radius is v, discard everything from that screen inward and retain only
the arm test starting at C_0 v, with its own screens (C_0 provides a
fixed unused gap). Quasimultiplicativity (A2) and the lower bound in
(A1) give, when C_0 v < s:
A(C_0 v, s) <= C (v/r)^g A(r, s).
Summing the failed-screen terms over geometric v costs at most
C A(r, s) sum_{v >= r} (v/r)^g exp(-c v^2) <= C A(r, s).
When C_0 v >= s, the exponential bound and the polynomial lower bound
give the same conclusion. The corresponding construction for one-arm
and straight half-plane three-arm probes preserves their displayed
upper bounds. These are called SCREENED ARM PROBES. Their supports can
be chosen at fixed-factor separations, and they can be used in the
presence of inserted points outside their buffers.
INPUT D (necessary arms for a local change). A color change or a local
insertion that switches a crossing requires four alternating arms out
of its interior support. Near a straight side of a quadrilateral, the
four arms continue as three alternating half-plane arms. Beyond the
scale of a corner, at least one arm continues. These are Lemma 2.2
of [V]; the statement uses the exact closed-cell crossing convention.
The same conclusion holds for a MIXED difference of two local changes:
if the mixed difference is nonzero, then in at least one of the four
configurations each local change affects the appropriate crossing, so
the four-alternating-arms requirement applies to that configuration.
Screens keep the other modification out of the arm probes used for
this implication.
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9. Scaling-limit inputs (from [V])
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Everything in this subsection is qualitative. The ORDER of limits will
matter throughout: all radii, buffer widths, and domain tests are fixed
BEFORE the mesh tends to zero.
INPUT E (scaling-limit inputs). Under the scalar Cardy Assumption:
(E-i) From every sequence of Poisson meshes tending to zero, one can
extract, after coupling, a further subsequence along which the
graphs almost surely form a deterministic good sequence in the
sense of [V, Lemma 2.5]. Along such a sequence, critical crossings
of Jordan quadrilaterals converge to Cardy's value, also for graph
domains whose boundaries and marked points converge in curve order.
Finite joint crossing tests and tests with finitely many well-
separated prescribed squares have the common triangular-percolation
limit.
(E-ii) Chordal color explorations in converging Jordan graph domains
converge to chordal SLE_6 (kappa = 6). At a pre-query stop on a
crosscut, the remaining target face, its boundary banks, and its
conformal maps converge in the sense of the stopped-domain results
of [V, Prop 3.1, Lemmas 3.4-3.7].
(E-iii) There is c_0 > 0 such that, in the mesh-first limit, the
probability of four bulk arms with any specified color order is
at most C (r/s)^(1 + c_0), and the probability of six bulk arms
with any specified color order is at most C (r/s)^(2 + c_0).
Three free arms at a regular boundary, measured in a conformal
chart, have an upper bound with exponent strictly greater than one
(in fact exponent 4/3 by [V, Lemma 2.11]).
"Free" arms means arms that use the independent fair interior sites,
not the sites in the prescribed base arc.
Precise locators: (E-i) uses Prop 2.4, Lemma 2.5, Prop 4.5 of [V];
(E-ii) uses Prop 3.1 and Lemmas 3.4-3.7; (E-iii) uses Lemmas 2.7, 2.8,
2.11 of [V].
TWO DOMAIN QUALIFICATIONS (we will be careful with both):
(Q1) A boundary chart must converge uniformly on its closure, have no
constant boundary subarc, and project injectively on the compact
bank interval where the estimate is used. Distinct prime-end banks
need NOT have disjoint physical projections. The local estimate
applies in an annulus by choosing a small graph disk patch with one
side on that bank and the remaining sides in the interior. These
are Jordan approximations; reflection in the parameter coordinate
gives bounded distortion on smaller patches.
(Q2) Product colors are conditional on the graph. After an exploration,
its remaining colors are fresh, but a remaining-domain theorem can
be used only AFTER its geometric hypotheses have been established.
We never infer uniform convergence after conditioning on arbitrary
atypical explored colors. Instead (Part VI) we use INDEPENDENT CORE
VARIABLES and bounds on the signed weights.
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10. Site triangulation and walls
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We use the site triangulation of the Voronoi tiling from [V].
Its faces are triangles, drawn THROUGH the tiling rather than required
to have straight edges. An interface strand, or WALL, passes through a
triangle between two heterochromatic edges. Inside a screened collar,
simple site cycles approximate circles in curve order. Such cycles are
constructed by following narrow strips and retaining simple cuts;
same-boundary chords on the domain-facing side can be shortcut while
retaining the specified inner region. Auxiliary uniform angular choices,
if needed, preserve rotation invariance.
Two facts about walls in a triangulation:
- Walls neither branch nor cross.
- Filling a disk bounded by a simple site cycle determines a
noncrossing pairing of the switches on the cycle by walls
inside the disk.
All transfer cuts in this paper lie in the BULK: their incident cells
and vertex stars stay away from the quad boundary, where site connectivity
and cell connectivity could otherwise require additional care. At the
actual quad boundary, we retain the original closed-cell event and use
the necessary-arm estimates of INPUT D.
Null nongeneric geometries, and null sets of insertion locations in
integrals, are discarded throughout.
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11. Calibration and the preliminary exponent
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Let
S_eps = P(a nucleus inserted at the center of Q_0 = [-1,1] x [-1/2,1/2]
is color-pivotal for the left-right crossing at
intensity eps^(-2)).
Equivalently, S_eps is the expectation of the black-forced crossing
indicator MINUS the white-forced crossing indicator for that site.
Define
m_eps = eps^(-2),
q_eps = (E N_eps(Q_0))^(-1),
f(R) = S_{1/R}.
The last definition agrees by Euclidean scaling with the unit-intensity
f(R) used in Part I.
INPUT F (calibration). There are constants c, B in (0, infinity) such that
m_eps * q_eps * S_eps -> c, (calibration-eq-a)
q_eps * E N_eps -> B. (calibration-eq-b)
Moreover,
f(R) \asymp A(b_*, R). (calibration-comparison)
In words: (a) says the single-cell pivotal probability times the number
of cells and its normalized weight is constant; (b) says the calibration
normalization converges to a positive constant; (comparison) says the
"insertion is pivotal" probability at unit intensity is comparable to
all four arms emanating from microscopic radius b_* to radius R.
These limits are the single-time Poisson specialization of [V, Lemmas 7.1
and 7.2] ("Common calibration" and "The prescribed normalization").
The signed color weight at the inserted site is (+1) - 0 = 1 for black
and 0 - 1 = -1 for white; crossing monotonicity identifies the expectation
with S_eps. The comparison (calibration-comparison) follows from the
local pivotal bounds and the square-to-site calibration in Sections 2
and 6 of [V].
INPUT F does NOT assert that R^(5/4) f(R) converges. That convergence is
what we are going to prove.
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12. The preliminary exponent, WITHOUT an amplitude
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LEMMA 12.1 (exponent without an amplitude). Under the scalar Cardy
Assumption,
f(R) = R^(-5/4 + o(1)) as R -> infinity. (exponent-eq)
PROOF. Force all cells meeting a central square of radius h in Q_0 first
black and then white. Let D_eps(h) denote the difference of crossing
probabilities. [V, Prop 4.5] gives, for each fixed small h > 0, a common
limit of this difference in the Voronoi and triangular models. The upper
arm bound (A1) together with [V, Lemma 6.1] ("A square can transmit a
pivotal change") compares D_eps(h) with the four-arm probability from
radius h to radius 1, with constants uniform in small h after taking
the mesh limit. The corresponding triangular four-arm probabilities
have exponent lambda = 5/4 by Smirnov-Werner.
Now fix any eta > 0. For an arbitrarily large but fixed ratio L = 1/h,
these comparisons and qualitative convergence imply, for all sufficiently
large r depending on L,
c_eta * L^(-lambda - eta/3) <= A(r, Lr)
<= C_eta * L^(-lambda + eta/3).
Choose L so large that both constants and the quasimultiplicativity
constants in (A2) are absorbed by an additional factor L^(eta/3). Only
AFTER fixing this L do we choose a starting radius above the convergence
threshold. Iterate over radii r, Lr, L^2 r, ... to get four-arm bounds
with exponents between lambda - eta and lambda + eta. The bounded initial
range and the final incomplete ratio change only the multiplicative
constants, by (A1) and (A2). Equation (calibration-comparison) then yields
C_eta^(-1) R^(-lambda - eta) <= f(R) <= C_eta R^(-lambda + eta).
Let eta -> 0. QED.
From now on we write lambda = 5/4. The remaining task is to replace the
o(1) in (exponent-eq) by convergence of the multiplicative coefficient.
The next two analytic Parts (III-IV) identify which annular symmetries
make that improvement possible.
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PART III. FOUR-INTERFACE TRANSMISSION IN AN ANNULUS
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What this Part delivers.
A conformal-limit object T^pm(p, x, y): the probability that four
interfaces cross an annulus of modulus p, given four-switch data on
each boundary, with an inner "cap" wiring black-to-black (+) or
white-to-white (-). We prove T^pm exists, is continuous, and has the
symmetries we need in Parts IV-VI. The proof uses the cap identity
(Lemma 13.1) which is a purely planar statement, plus the scalar
Cardy hypothesis via the input E.
Why this Part is not a direct application of Cardy.
Cardy's formula applies to Jordan QUADRILATERALS: simply connected
domains with four marked boundary points. An annulus with four
switches on each boundary is multiply connected. Ordinary Cardy
does not apply. We must construct four-interface transmission from
simply connected pieces.
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13. Connection states, walls, and the two caps
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We work with the site triangulation of the Voronoi tiling from [V].
Its faces are triangles drawn through the tiling; walls are interface
strands between heterochromatic edges.
Definitions.
A RUN on a simple site cycle is a maximal consecutive set of sites
of one color. A SWITCH is the edge between consecutive runs.
A CONNECTION STATE on the cycle is a noncrossing pairing of the
switches, together with the color pattern of runs. This pairing is
determined by filling the disk bounded by the cycle and drawing the
walls inside. Closed walls (loops entirely inside the disk) are not
recorded: they carry no boundary connectivity information.
Equivalently the run colors and the pairing together determine
exactly which boundary sites are connected through the interior.
Suppose now we have a graph annulus whose two boundary components each
have four switches and four alternating prescribed runs (black, white,
black, white). All other sites in the annulus carry independent fair
colors. Let
T = { all four outer wall ends reach the inner boundary }.
By noncrossing property of walls, if T fails (i.e. fewer than four
inner ends are connected to outer ends) the pairing visible from the
outer cycle is unaffected by which cap we place on the inner cycle.
We place a "cap" on the inner cycle: a formal disk filling the inner
cycle that joins the two black runs (this is the BLACK cap, state B)
or joins the two white runs (this is the WHITE cap, state W).
These are FORMAL planar caps: we do not sample extra random colors
in them. The cap is a deterministic gadget that decides connectivity
of the inner boundary sites.
Let
H_b = 1_{the two outer black runs are connected after the
black cap is attached to the inner cycle},
H_w = 1_{the two outer black runs are connected after the
white cap is attached to the inner cycle}.
Note: both H_b and H_w refer to connectivity of the OUTER black runs.
They differ in what we do on the INNER cycle.
LEMMA 13.1 (the cap identity). For every coloring of the annulus,
H_b - H_w = 1_T. (cap-equation)
Consequently, the signed difference between the two inner cap states
is transported to the same signed difference at the outer cycle,
multiplied by the indicator of four-interface transmission. The identity
also holds under composition across several cycles.
PROOF. Walls in a triangulation neither branch nor cross.
Case A: fewer than four inner ends connected to outer ends (i.e. T fails).
Then the wall pairing visible from the outer cycle involves zero or two
surviving inner ends; the inner cap has no effect on how four-interface
connectivity between outer black runs either succeeds or fails. Hence
H_b = H_w, so H_b - H_w = 0 = 1_T.
Case B: all four inner ends connected to outer ends (T holds).
The cyclic order of the four wall arcs is preserved. The bands between
them are monochromatic; they carry black connectivity to black and
white to white. Attaching the black cap (state B) makes the outer black
runs connected through the two black inner runs, so H_b = 1. Attaching
the white cap (state W) instead makes the outer white runs connected
through the two white inner runs, which prevents the outer black runs
from being connected through the interior; the four outer black runs
are the two ends of distinct black bands that do not touch through a
white cap; hence H_w = 0. Thus H_b - H_w = 1 = 1_T.
Hex duality after contracting each run gives the same planar argument.
Composing the planar pairings (gluing the outer cap of one annulus
to the inner cap of another) preserves the identity; closed walls
(introduced by the composition) trivially have weight one. QED.
Remarks.
- For fixed coloring, H_b - H_w takes values in {0, 1} and equals
the indicator that all four walls are transmitted.
- Under composition of two glued annuli, the "cap difference" at the
far end is the same as at the near end, multiplied by the
transmission indicator of the intervening annulus.
- The identity does not require any Poisson randomness in the caps:
the caps are formal planar choices, and H_b, H_w are deterministic
functions of the coloring given the choice of cap.
----------------------------------------------------------------
14. Boundary conventions for limit statements
----------------------------------------------------------------
For p > 0 let
A_p = { z : e^(-p) < |z| < 1 }.
Thus p = log(R/r) for a round annulus with radii r < R (the standard
logarithmic modulus). We parametrize both circles by angle in [0, 2pi)
with counterclockwise orientation.
x = (x_1, x_2, x_3, x_4) = outer switch angles,
y = (y_1, y_2, y_3, y_4) = inner switch angles,
using INCREASING lifts with x_4 < x_1 + 2pi and y_4 < y_1 + 2pi.
Translation of either lift by 2pi does not change the data. The run
following the first outer switch is black; for the "+" prescription
the corresponding inner run is black, and for the "-" prescription it
is white.
All limits in this Part are taken along deterministic good sequences of
triangulations in the sense of [V, Section 2] ("Deterministic sequences
and Jordan tests"). That is: the mesh tends to zero on compact sets;
joint Jordan-quad crossing limits hold; and the mesh-first arm estimates
are available. Probabilities are over product colors conditional on
these triangulations.
----------------------------------------------------------------
15. Admissible boundary convergence
----------------------------------------------------------------
DEFINITION 15.1. A sequence of marked graph annuli D_n has ADMISSIBLE
BOUNDARY CONVERGENCE if it lies in a fixed bounded set and satisfies:
(C1) Each boundary is a simple site cycle. With rotations normalized,
the uniformization phi_n : A_{p_n} -> D_n extends continuously
to the closed annulus, with p_n -> p in (0, infinity). After
linear identification in logarithmic radii, phi_n converge
uniformly on the closed annulus to a conformal, one-to-one
phi : A_p -> D. Boundary traversals give prime-end accesses.
(C2) Each limiting boundary traversal is a concatenation of finitely
many simple arcs, each with its own access. Different arcs may
meet in physical projection. The two boundary components remain
distinct as prime-end boundary components.
(C3) The four switches on each component converge in these coordinates
to four distinct switches. Between them the prescribed run colors
alternate. The sites strictly between the two cycles have fresh
independent fair colors.
At graph level we take boundaries without chords facing the domain
when that is required for a local boundary chart. Such cuts come from
the graph approximation and shortcut construction of [V, Section 2];
at an exploration frontier, edges between already revealed vertices
are included before the frontier is taken.
The definition permits a slit to have two DIFFERENT banks, even when
their images coincide. It does not identify these banks. This is exactly
what we need when a stopped exploration produces a boundary that looks
like a self-touching curve.
----------------------------------------------------------------
16. Gluing a back with finitely many attached lobes
----------------------------------------------------------------
We will meet the following situation many times: an annulus consists
of an intact annular piece plus finitely many Jordan "lobes" glued on
along open intervals of a boundary. When do the annular
uniformizations of such domains converge?
LEMMA 16.1 (back with finitely many attached lobes). Suppose graph
annuli consist of an open annular back and finitely many Jordan lobes
attached to it along open seam intervals. Assume:
(G1) The backs are Jordan annuli whose two complete boundary
traversals converge in curve order to those of a fixed Jordan
annulus.
(G2) The seam endpoints converge in that order, and the limiting
seams remain open.
(G3) Each lobe converges as a Jordan domain with its boundary traversal.
(G4) Seam incidences and cyclic order are fixed. Lobes are glued only
along their specified seams; their interiors are disjoint from
the back and from one another.
(G5) The limiting domain is an annulus; its hole and a separating
open part of the back persist; the two boundary components stay
physically apart.
(G6) Different limiting lobes may touch in projection but have
disjoint interiors.
Then the annular uniformizations of the whole domains converge
uniformly on their closures, with the stated accesses and traversal
order. If the marked accesses converge and remain distinct, we get
admissible boundary convergence (Definition 15.1).
PROOF.
Step 1 (kernel convergence with accesses). Every compact subset of a
lobe persists by Jordan convergence. A lobe joins the back across
every compact subinterval of its open seam. Conversely, every point of
the limiting boundary is approached by the approximating boundaries.
Contacts between different lobes do not open passages: the individual
Jordan pieces are attached along their specified seams, not through
a point of contact. These three facts together give kernel convergence
with the indicated accesses.
Step 2 (modulus convergence). The persisting hole and separating back
preclude degeneration of the modulus. The conformal kernel theorem
therefore gives convergence of normalized annular maps on interiors
and convergence of their moduli.
Step 3 (boundary parametrization). Fix a nonconstant open subarc of
one of the finitely many Jordan boundary pieces and a point in the
persisting back. Jordan-map convergence in a lobe (or in a disk patch
along an unchanged back boundary) gives a positive lower bound for the
harmonic measure of a smaller subarc as seen from a point in that patch.
A Harnack chain through persisting interiors gives the same lower
bound from the fixed back point. In round annular coordinates the
harmonic-measure densities from that preimage are bounded uniformly
above (moduli and preimage stay in compact nondegenerate ranges).
Consequently no fixed nonconstant subarc of the boundary traversal
can be squeezed into an angular interval of vanishing length.
Step 4 (equicontinuity). Jordan convergence of the finitely many
pieces gives convergence of the full boundary traversals in curve
order. Partition the traversals into finitely many small image arcs;
apply the preceding lower bound to intervening nonconstant subarcs.
If equicontinuity in angular parameters failed, one fixed subarc would
be traversed in a vanishing angular interval, contradicting Step 3.
Step 5 (identification of limits). Interior convergence identifies
every uniformly convergent subsequence. Small crosscuts at the finitely
many joins extend the conclusion to the joins. There is no constant
boundary interval in the limiting conformal map (uniqueness of a
holomorphic function continuous on a boundary arc). The boundary order
and converging marked accesses thus identify the limiting switches.
QED.
----------------------------------------------------------------
17. Boundary motion: continuity from removed roughness
----------------------------------------------------------------
The rough banks of an admissible annulus make the "crossing" notion
non-Jordan. We first prove that we may MOVE each boundary slightly
INWARD the annulus without changing the crossing event, up to vanishing
probability. After this lemma we may assume banks lie on smooth contours
strictly inside the parameter annulus, with spare space on both sides.
The arm bounds we use in this lemma are mesh-first statements:
three free arms along one regular bank have an upper bound with exponent
1 + eta for some eta > 0; a free arm from a shrinking neighborhood to
a fixed scale has probability tending to zero.
LEMMA 17.1 (continuity under boundary motion). For a sequence with
admissible boundary convergence, move either cycle slightly into the
annulus in round coordinates, and move its switches slightly with it.
Couple all unchanged free colors. The probability that either capped
connection test changes tends to zero as the size of the motion tends
to zero, after taking the mesh limit. The assertion also holds for
small piecewise smooth motions in the spare space around an inset
annulus. It is uniform along any convergent sequence of admissible
data with distinct limiting switches.
More precisely, if H_n and H_{n, delta} are either capped indicators
BEFORE and AFTER a motion of size at most delta, preserving run order
and labels away from the moving switches, then
lim_{delta -> 0} limsup_{n -> infinity}
P( H_n != H_{n, delta} ) = 0. (motion-limit)
PROOF.
Step 1 (mesh in annular coordinates). First observe that the mesh in
annular coordinates tends to zero. If not, a connected cell or triangle
portion of vanishing physical diameter would have preimages with a
nontrivial continuum limit on which phi is constant. But phi is one-to-one
on the interior and has no constant boundary interval (Definition 15.1
C2), so such a continuum is impossible. Portions of the GRAPH (not whole
cells crossing the domain boundary) are used here.
Step 2 (local patches). Remove small coordinate neighborhoods of the
switches and of the finitely many joins of simple bank arcs. On each
remaining compact bank interval, use disk patches whose boundary consists
of that bank plus cuts strictly inside the annulus. The physical boundaries
are Jordan and the graph approximations converge in curve order. Their
uniformizing coordinates agree (with fixed-scale margins) with the
annular coordinates. Reflection in the limiting round bank controls
distortion on smaller patches. Thus the local boundary-arm estimates of
[V] apply on these patches even when another prime-end bank has the
same physical image.
Step 3 (motion as a sequence of local replacements). Cover these intervals
by O(1/s) coordinate disks of radius s. Implement a motion much smaller
than s by successively replacing bank intervals, with tapered ends. At
graph level choose short transverse joins in disjoint narrow strips from
the old cycle to its replacement. Substitute intervals in cyclic order,
retaining the old cycle off the interval currently being changed, and
erasing the joins once the whole cycle is replaced. The graph
approximation in curve order makes these simple cuts. Each join has the
same constant color as its run. Moves containing switches or bank joins
are handled inside the exceptional neighborhoods above. Two nearby
nonnested contours can be compared through a common slightly inset
contour.
Step 4 (three arms if a capped test changes). Consider an ordinary move
on a black bank. Suppose a capped connection test changes. Choose a
winning black connection in one test and a winning dual white connection
in the other. Contract the prescribed runs; use the same cap in both
tests. Purely wired connections do not change, so a free segment of a
winning path must meet the support of the move. The test neighborhood
avoids every other run and mark. The white path has two free branches
leaving the neighborhood. At least one black branch must also leave
freely: a black branch already reaching the same prescribed run outside
the moved interval could be joined there without the local change. After
contraction, the branches are disjoint. Two white branches plus the
opposite-colored black branch give three disjoint free arms in a
slightly shifted half-patch, with fixed-factor margins. Forced sites on
intermediate cuts are not counted as free arms.
Step 5 (cost of the change). For any fixed larger chart scale, the
mesh-first bound for a single disk is O(s^(1 + eta)). The union over
O(1/s) disks therefore costs O(s^eta). The number of exceptional
neighborhoods is finite and fixed. A change there requires a fair arm
approaching from positive parameter distance (connections already
supplied by the same run or the cap do not change). This arm has
positive physical diameter uniformly (by Step 1). The ordinary physical
one-arm bound (A3) makes the exceptional cost tend to zero, including
at a contact between different accesses. We may move each exceptional
neighborhood as a single piece, so this cost is not multiplied by the
number of ordinary intervals.
Step 6 (order of limits). Take mesh limit first; make ordinary motions
small vs s next; let s -> 0; finally shrink exceptional neighborhoods.
This proves the assertion. The proof is identical with moving marks
or a moving round radius. QED.
After Lemma 17.1, all prescribed banks may be placed on smooth contours
strictly inside the parameter annulus, with spare space on both sides.
There are two further issues:
(i) joint ordinary crossings at conformally smooth banks must have a
conformally invariant annular law;
(ii) those ordinary crossings must determine the capped tests.
We address them in the next two subsections, in that order.
----------------------------------------------------------------
18. Ordinary crossings in annular open sets are conformally invariant
----------------------------------------------------------------
Before we can talk about T^pm as a conformal object we must know that
ordinary crossing events (no switching, no cap: just black or white
crossings of quadrilateral tubes) have a well-defined joint limiting
law that is invariant under conformal maps of the annulus.
LEMMA 18.1 (ordinary crossings in an annular open set). In an annular
open set compactly contained inside the parameter annulus, the joint
limiting law of any countable collection of regular ordinary quad tests
is conformally invariant. The tests may be chosen from a countable
dense family of polygon or piecewise smooth quads with positive margins.
PROOF.
Step 1 (per disk). On every strict Jordan disk patch the joint law is
the critical triangular percolation law by [V, Sections 3-4]. This law
is conformally invariant (Camia-Newman Theorems 1 and 7; quad-crossing
formulation in Garban-Pete-Schramm Section 2.3). Scalar Cardy sandwiches
identify smooth images of polygon tests and nearby graph crossing
conventions with the same limit.
Step 2 (finite collection off a cut). Choose a finite-length smooth cut
opening the annulus to a disk. Any finite collection of tests compactly
contained off the cut fits in a strict Jordan disk, so its joint law
agrees in the physical domain and in the conformal chart.
Step 3 (recovering tests meeting the cut). To recover tests meeting
the cut, use the Schramm-Smirnov gluing theorem only in strict Jordan
disk patches (Sections 1.19 and 4.1 of their paper). For the critical
triangular crossing law, the crossing of a regular quad in such a patch
is measurable, modulo null sets, with respect to countably many tests
avoiding the finite smooth cut in that patch. Its hypotheses hold here:
independence in the discrete model, RSW, one-arm decay, and a four-arm
exponent strictly greater than one. Regular continuity tests may be used
throughout by Cardy sandwiches.
Step 4 (transfer to the chart). Each required quad meeting the annular
cut has a slightly larger Jordan disk neighborhood inside the annulus.
The recovering measurable rule in that neighborhood is the same in the
chart and in its image, since the whole disk laws already agree. All
recovering tests belong to a common countable off-cut family. Hence
the recovered crossings have the same joint law as well.
Step 5 (duality). White crossings are included by Hex duality and
continuity. No gluing assertion along a rough slit is used; only along
smooth cuts in strict Jordan disk patches. QED.
----------------------------------------------------------------
19. The annular transmission limit
----------------------------------------------------------------
Now we can define T^pm as a limit.
PROPOSITION 19.1 (annular transmission limit). For every sequence with
admissible boundary convergence, the probability of T (four interfaces
transmit) has a limit depending only on the round data (p, x, y) and
the choice of inner colors. Denote these limits by
T^+(p, x, y), T^-(p, x, y).
They are continuous for p > 0 and distinct switches.
The same convergence holds for each capped connection probability.
In particular, it applies to random remaining domains whenever
- conditional on their geometry and explored boundary, their interior
colors are fresh, and
- their marked domains have admissible convergence almost surely in a
subsequential coupling.
In that case the difference between the conditional transmission
probability and T^pm evaluated at the actual annular conformal data
tends to zero in mean absolute value.
PROOF.
Step 1 (smooth inset reduction). By Lemma 17.1, it suffices to consider
smooth inset annuli with spare space. We show that ordinary quad
crossings determine the limiting value of each capped black connection
test.
Step 2 (common coupling). Sample fair colors throughout a slightly
larger annular neighborhood in the spare space, overriding them on the
prescribed runs only when evaluating a capped test. Ordinary tube tests
use the fair colors. Compare two candidate sequences, one of which may
be in round coordinates. Couple their countable ordinary crossing vectors
through the common limiting law of Lemma 18.1. Along further subsequences
include the cap indicators in the coupling; the curve tightness proved
below also allows the chosen winning paths to be included. The two
limiting collections are coupled to have the SAME ordinary crossing
vector.
Step 3 (favorable perturbation). Fix a small FAVORABLE perturbation of
the black test in the second sequence. Move black banks into the annular
fluid and extend their runs slightly past the old switches; retreat white
banks from the old fluid. Use positive slacks and taper the motions
within the old white intervals, putting each new switch on the retreated
part. Then: every old black attachment lies behind a new black bank; new
white banks are positively separated from the old fluid; removed fluid
is in separate collars of the expanded black runs. All choices are in
the spare space.
Step 4 (winning path). On a positive configuration in the first sequence,
contract each prescribed run and attach the cap edge when black. Take an
outermost simple winning path between the two outer black terminals (for
instance, attach an external terminal edge and take the appropriate
boundary of the union of cycles containing it). Contracting connected
boundary runs and attaching the cap preserves the planar embedding. The
resulting path has NO BYPASS on its chosen exterior side. It visits each
contracted run at most once, so it decomposes into a bounded number of
free black segments joining distinct runs, together with wired or cap
joins. Crop their endpoints by a vanishing graph distance if they end
adjacent to a wire.
Step 5 (tightness of free segments). Many oscillations across an
annulus supply many disjoint free arms; BK and one-arm decay bound
their probabilities. Successive-excursion compactness gives tightness
in the curve metric. A bulk self-contact with two positive excursions
would give four black traversals. In two gaps on the no-bypass side,
intermediate transverse cuts and Hex duality give two additional white
traversals: a black transverse crossing would instead be a bypass. With
fixed-factor crops the six arms are disjoint, also for backtracking
pieces. The mesh-first six-arm exponent greater than two (E-iii)
excludes such contacts by a covering argument. At a smooth boundary
bank, a repeated visit to the same point gives at least three free
traversals; the boundary exponent greater than one (E-iii) excludes
these repeated visits. A single tangential bank contact need not be
excluded: the favorable perturbation accommodates it. At the finitely
many switches one free arm suffices. Constant pauses may be removed.
This is exactly the local no-bypass and free-segment argument of
[V, Section 4].
Step 6 (transplanting a winning path). In a subsequential coupled
representation, fix the favorable displacement. Crop each limiting
free segment a small positive amount at its ends, keeping new endpoints
strictly behind the expanded black banks. (A cropped endpoint may still
lie on an old bank; the ambient tube may straddle that bank within the
spare space. This positive crop is distinct from the vanishing graph
crop used to remove incidences to a wire.) Surround the cropped simple
segment by a thin Jordan tube with end caps in the endpoint neighborhoods.
The approximating free graph segments supply ordinary fair-color crossings
of this ambient tube. The tube can be approximated with positive margins
by a quad from the countable ordinary family. Its crossing occurs in the
first sequence, and hence in the second for all sufficiently large
indices in the coupling. Clip these transferred crossings when they hit
the new black banks. Excursions outside the new fluid can only lie in
the collars now wired black; short them through the corresponding run
(collars of different black banks are separated). At graph level every
passage between a removed collar and the fluid meets its boundary cycle.
The transferred tubes, together with the unchanged cap or wire joins,
give a winning black connection in the favorably perturbed second test.
Step 7 (finish). Apply Step 6 to subsequences realizing the upper and
lower limits of the unperturbed cap probabilities. The upper limit of the
first is bounded by the lower limit of the second plus its boundary-motion
error. Remove the favorable perturbations using Lemma 17.1, reverse the
roles of the sequences. Their upper and lower limits coincide. This proves
convergence and conformal invariance of each cap test on inset annuli.
Take the inset size to zero to get the same assertions under Definition
15.1 (admissibility). Cap identity gives the transmission limit. Allowing
marks and round radii to move (same argument) proves continuity in
(p, x, y).
Step 8 (random remaining domains). For almost every convergent domain
realization, the deterministic statement applies to its fresh
product-color law; continuity identifies the same limit for T^pm at the
actual conformal data. Probabilities are bounded by one, so bounded
convergence gives convergence in mean absolute value. QED.
A note on what Proposition 19.1 does NOT say. It does not assert a
uniform conditional estimate over arbitrary atypical explored boundaries.
Only the mean-absolute-value statement is made, and only along the
stated coupling.
----------------------------------------------------------------
20. Behavior at collisions and symmetries
----------------------------------------------------------------
We now record the boundary behavior of T^pm and the symmetries we need.
LEMMA 20.1 (collisions and symmetries). The functions T^pm extend
continuously by zero when switches collide on either boundary, uniformly
for p in compact subsets of (0, infinity). They are invariant under
- simultaneous rotation of x and y,
- global color reversal (interchanging black and white),
and cyclic relabeling by one switch in either tuple interchanges T^+
and T^-.
PROOF. Suppose two consecutive inner (or outer) switches coalesce. Four-
interface transmission requires a free arm from their intervening
shrinking run to the OTHER boundary. This free arm cannot use any other
prescribed run: it lies between the two corresponding transmitted walls.
A mesh-first one-arm bound with fixed outer scale and shrinking inner
scale gives vanishing. The bound does not depend on the positions of
the other switches and therefore covers simultaneous collisions.
Conformal invariance (Proposition 19.1) gives simultaneous rotational
invariance, and fair colors give global color symmetry. Moving the first
switch to the next switch reverses the alternating run labels on that
component, which by the defining convention interchanges the two inner
prescriptions (and hence T^+ and T^-). QED.
================================================================
PART IV. LARGE-MODULUS ASYMPTOTICS OF THE
FOUR-INTERFACE TRANSMISSION
================================================================
What this Part delivers.
(IV-a) The rank-one asymptotic:
T^pm(p, x, y) = e^(-5p/4) * (K h(x, y) + O(e^(-sigma p)))
with K in [0, infinity), h(x, y) = J(x)^(1/3) J(y)^(1/3),
and sigma > 0.
(IV-b) The SIGNED asymptotic with rotation average:
||W_-(p)|| <= C e^(-3p/4)
||Pi W_-(p)|| <= C (1 + p) e^(-p)
where W_- = e^(5p/4)(T^+ - T^-)/h and Pi is the average over
simultaneous rotation of one end.
The first inequality already gives exponent 3/4 > 1/2;
the second gives exponent 1 < 2 - sigma. We will need the
stronger exponent > 2 in Part V (rotation average of signed
T^+ - T^- decays like e^(-9p/4) which survives composition).
Strategy. The annular transmission function T^pm satisfies a parabolic
PDE in the modulus p and the switch angles x, y. This PDE is the
generator of an annulus SLE_6 exploration (Zhan) traced from FOUR
independent active switches at once. Understand the PDE spectrally,
and the asymptotics follow.
----------------------------------------------------------------
21. Modulus space and notation
----------------------------------------------------------------
The space of four cyclically ordered, LABELED angles is
C = { w = (w_1, w_2, w_3, w_4) in R^4 :
w_1 < w_2 < w_3 < w_4 < w_1 + 2pi } / (2pi Z * 1).
We quotient by simultaneous full turns, not by cyclic relabeling.
We work on the product chamber
X = C cross C (coordinate (x, y)).
All statements about distances or derivatives use local angle lifts.
Constants are uniform as angles approach a collision, and moduli are
bounded below by a fixed positive constant.
----------------------------------------------------------------
22. The modulus equation
----------------------------------------------------------------
PROPOSITION 22.1 (modulus equation). The functions T^pm are smooth on
(0, infinity) cross X and satisfy
partial_p T = ( L + B_p . grad ) T, (pde)
where
L = (1/8) sum_{w = x, y} [ 3 sum_{i=1}^4 partial_{w_i w_i}
+ sum_{i=1}^4 sum_{j != i}
cot((w_i - w_j)/2) partial_{w_i} ].
For every p_0 > 0 and every finite derivative order k, the vector field
B_p extends smoothly across angle collisions and
|| B_p ||_{C^k} <= C_{k, p_0} e^(-p) (p >= p_0).
B_p is equivariant under permutations within either tuple, common
rotations, and cyclic relabelings. In particular, its NORMAL component
at every collision face is zero.
PROOF.
Step 1 (local equation with one active switch). Write the annulus as
the cylinder 0 < Im z < p, with real period 2pi. Take the active switch
on its lower boundary. Parametrize its exploration by the decrease of
the modulus. In the annulus Loewner normalization (zero regular constant
at the driving pole), the angular kernel is
H_p(u) = lim_{N -> infinity} sum_{k = -N}^N cot((u - 2 i k p) / 2).
The pole has residue 2, and the imaginary part on the other boundary is
-1. The locality theorem for annulus SLE_6 (Zhan) identifies the local
driving angle with Brownian motion of variance parameter 6. Only locality
BEFORE leaving a small simply connected neighborhood of the active
switch is used here.
Step 2 (why no extra drift). Absence of an additional driving drift
in the LOCAL equation depends on the normalization. In a chordal chart
the pole part is 2/(z - U_t). If F_t is a changing local conformal
coordinate, comparison of pole residues gives the new clock
F_t'(U_t)^2 dt. Comparison of the regular terms, both normalized to
have zero constant, gives partial_t F_t(U_t) = -3 F_t''(U_t). This
cancels the Ito correction kappa F_t''(U_t) / 2 precisely for
kappa = 6. No condition on a remote target in the annulus enters this
calculation.
Step 3 (other switches). Other switches on the active boundary move
with velocity H_p(u), where u is their angular difference from the
active one. Switches on the opposite boundary move with angular
velocity Re H_p(u + i p). Pairing positive and negative terms of the
series gives, uniformly with any fixed number of derivatives:
H_p(u) - cot(u/2) = O(e^(-2p)), (kernel-errors)
Re H_p(u + i p) = O(e^(-p)).
The remainders are smooth periodic functions of u, including at u = 0.
Step 4 (active equations). When x_i is the active switch, the equation
for T is
0 = - partial_p T + 3 partial_{x_i x_i} T
+ sum_{j != i} H_p(x_j - x_i) partial_{x_j} T
+ sum_{j = 1}^4 Re H_p(y_j - x_i + i p) partial_{y_j} T.
(active-equation)
For an active switch y_i, exchange x and y. (The cylinder change
z -> -z + i p exchanges boundaries and reverses both angle coordinates;
the angular kernels are odd, so transforming back preserves the displayed
signs.)
Step 5 (average eight active equations). Average the four equations
from x_1,...,x_4 active and the four equations from y_1,...,y_4 active.
This gives (pde). The series estimates give the bounds on B_p. The
averaged expression treats the labels within each tuple symmetrically,
so it is permutation-equivariant. At a collision the two corresponding
components of B_p agree, giving tangency to the collision face. The
other symmetries follow the same way from the kernels.
Step 6 (Markov identity from stopped exploration). We justify the
Markov identity underlying these equations BEFORE assuming
differentiability. On a round annulus choose an arbitrarily small
crosscut around the active switch, separated from all other switches
and from the other boundary component. Approximate the smooth bank by
chord-free graph boundaries and complete the neighborhood to a Jordan
patch, putting its second color switch remotely. Up to the pre-query
stop at the crosscut, the exploration in this patch is EXACTLY the
exploration in the annulus: no color beyond the crosscut is read.
The stopped-exploration results of [V, Lemmas 3.6-3.7] give convergence
of the path together with its remaining domains. Crosscuts may be chosen
as semicircle crosscuts in the stopped-target construction; the
probability of an endpoint stop vanishes when endpoint neighborhoods
shrink.
The unaltered part, obtained by cutting off the fixed crosscut
neighborhood, is a Jordan annular back with unchanged boundary
traversal. For the remaining face containing this back, the stopped-
target-face and remaining-domain statements of [V, Lemmas 3.6-3.7]
give finitely many Jordan side pieces attached to the back along open
subintervals of that crosscut. Components already disconnected from
this face are discarded; no assertion of finiteness is made about all
components swallowed by the exploration.
We check that the limit is still doubly connected (Lemma 16.1
requirements). In the closed crosscut neighborhood, remove the side-lobe
interiors and their open attaching intervals. The remaining closed set
is connected to the original outer complement: join any of its points
to the old outer bank by a path whose interior avoids the crosscut; if
this path enters a lobe, replace its portion from first entry to last
exit by the corresponding subarc of the lobe boundary off its open
attachment. This subarc meets the crosscut only at its own endpoints
and enters no other lobe interior. Finitely many replacements give a
path in the complement. The unchanged inner hole is separated from this
connected outer complement by the persisting annular back. These are
exactly the two nondegenerate complementary components. Lemma 16.1
gives convergence of annular uniformizations on their closures, with
prime-end traversals and marks in order. The hole and a separating open
bulk persist, so the moduli remain nondegenerate. The residual annuli
satisfy Proposition 19.1 (transmission limit).
The stopping edge becomes the replacement switch; other marks are
unchanged; and colors in the remaining face are fresh. Transmission
topology plus bounded convergence give the local mean-value identity
for T^pm at each of these hull stops.
Step 7 (viscosity to classical). This identity first implies the active
equations in the VISCOSITY sense. Work modulo simultaneous rotation and
let a smooth test function touch T from above or below. On a sufficiently
small crosscut stop, the modulus and image marks remain in the touching
neighborhood: the hull capacity tends to zero, and annular maps
normalized at a remote mark tend to the identity on the other compact
bank intervals (by reflection). The image of the tip is squeezed between
neighboring unaffected bank points. Applying Ito's formula to the
touching function contradicts a strict inequality with the wrong sign
in the active equation, because the stop takes positive time. The same
touching function works for all eight active equations, so their average
also holds in viscosity form.
On every compact subset of the open chamber the averaged equation is a
smooth, uniformly parabolic LINEAR equation, also after the rotation
quotient. Interior parabolic regularity then gives smoothness and the
classical equation. Simultaneous rotation invariance removes the
quotient. QED.
----------------------------------------------------------------
23. The ground-state transform
----------------------------------------------------------------
Define positive functions on the open chamber:
J(w) = product_{1 <= i < j <= 4} sin((w_j - w_i) / 2),
h(x,y) = J(x)^(1/3) J(y)^(1/3),
lambda = 5/4. (ground-state)
The product J is unchanged by cyclic relabeling. For one tuple, put
s_i = sum_{j != i} cot((w_i - w_j) / 2).
The three-angle cotangent identity gives
sum_i s_i^2 = sum_{i != j} csc^2((w_i - w_j)/2) - 20,
partial_{w_i} log J^(1/3) = s_i / 6.
Substitution in L shows that each tuple contributes -5/8 to h^(-1) L h,
so
L h = - lambda h. (eigenfunction)
(This is the cotangent-diffusion form of the SLE / Calogero-Sutherland
structure; see Cardy's paper. We give the estimates needed here directly.)
The transformed generator and its invariant density:
D = (3/8) ( Delta + (grad log m) . grad ),
m(x,y) = J(x)^(4/3) J(y)^(4/3). (doob-generator)
Let mu be the probability measure proportional to m times Lebesgue
angle measure. For T = T^+, T^-, or a linear combination of them, set
W(p) = e^(lambda p) T(p) / h.
Equation (pde) becomes
partial_p W = D W + B_p . grad W + V_p W,
V_p = B_p . grad log h. (transformed-pde)
The apparent singularities of V_p at collisions are removable: every
cotangent is multiplied by a difference of two components of B_p, which
vanishes on the corresponding collision face. Periodicity handles the
face w_4 = w_1 + 2pi. So V_p has smooth bounds O(e^(-p)) up to the
chamber boundary, as does B_p.
----------------------------------------------------------------
24. Smoothing on the chamber
----------------------------------------------------------------
LEMMA 24.1 (smoothing on the chamber). Fix p_0 > 0. For either
transmission function, or its sum or difference, the transformed
solution satisfies
||W(p)||_infty + Lip(W(p)) <= C_{p_0} (p >= p_0 + 2).
(uniform-smoothing)
The semigroup P_t of D is symmetric on L^2(mu) and, for every q > 1,
maps L^q(mu) to bounded functions at every fixed positive time. At unit
time the evolution in (transformed-pde) maps bounded functions to
Lipschitz functions with a constant uniform in the starting modulus
p >= p_0.
PROOF.
Step 1 (why the diffusion runs to the chamber boundary). The drift in
one tuple is
(1/4) ( sum_{j != i} cot((w_i - w_j)/2) )_{i=1}^4.
On the lifted convex chamber, use -log m shifted to be nonnegative, as a
Lyapunov function. With alpha = 4/3 the density exponent, the cotangent
identity shows that its generator has a singular contribution
proportional to
(alpha - alpha^2) sum_{i < j} csc^2((w_i - w_j)/2).
Since alpha - alpha^2 < 0, the generator is bounded above. More
explicitly, the one-tuple contribution is
D(-log J^(4/3)) = 10/3 - (1/24) sum_{i != j} csc^2((w_i - w_j)/2).
Stopping on inner exhaustions proves that no collision occurs in finite
time. Interior existence and pathwise uniqueness give a global diffusion.
A bounded added drift, such as the time-dependent B_p, preserves
noncollision by Girsanov's theorem on each bounded time interval. The
divergence form of D gives symmetry and invariance of mu, first on
interior exhaustions and then by noncollision.
Step 2 (Feynman-Kac representation). The Feynman-Kac representation
for (transformed-pde) uses D + B . grad and potential V, with
coefficients read at DECREASING moduli from the evaluation modulus down
to the initial modulus. Its initial data at p_0 obey |W(p_0)| <= C / h.
To justify this representation despite the unbounded transformed data,
first apply the killed representation to T on inner chamber exhaustions.
Lemma 20.1 (collision behaviour) says the lateral values of T tend
uniformly to zero on each compact modulus interval, so their contribution
disappears. The h-transform on the exhaustion gives the stated
representation; pass to the limit using noncollision. It suffices to do
this for the nonnegative T^pm and then use linearity.
Step 3 (uniform L^q control). The negative logarithm of m is convex on
the lifted chamber, so the drift of D is monotone dissipative. Couple
two diffusions by the same Brownian motion and add to the second a drift
toward the first of magnitude their initial distance divided by the
allotted time, until they meet. Dissipativity forces meeting by that time.
Girsanov removes the extra drift with all fixed moments bounded in terms
of the time and initial distance. Localization inside the chamber
justifies the coupling, and the preceding change-of-measure argument
excludes collisions for the controlled diffusion. Holder's inequality
gives the dimension-free form
|P_t F(z)|^q <= C(q, t, dist(z, z')) P_t(|F|^q)(z').
Any two points z, z' admit lifts at bounded distance. Integration in z'
against the invariant probability mu proves the L^q-to-supremum
estimate.
Step 4 (initial smoothing). A bounded drift perturbation and a bounded
potential on a unit time interval preserve initial smoothing from
L^4(mu). Cauchy-Schwarz applied to the Girsanov density bounds the
perturbed expectation of |f| by C (P_1 |f|^2)^(1/2); the unperturbed
L^2-to-sup bound for |f|^2 then gives C ||f||_{L^4(mu)}. There is
exactly this integrability here: h^(-1) in L^4(mu) since
h^(-4) m = 1. After the first unit interval W is bounded. Its bound
persists for all subsequent times because
integral_{p_0}^infty ||V_p||_infty dp < infinity.
Step 5 (Lipschitz). The same coupling proves a Lipschitz estimate for
bounded data on a unit interval. In the presence of B, a control of size
at most a fixed constant times the initial distance forces meeting;
before meeting the distance stays bounded by the same multiple of the
initial distance. The second moment estimate for the Girsanov
exponential bounds its total-variation cost by that distance. The two
Feynman-Kac weights differ by at most the same order, since V has a
uniform Lipschitz bound. Thus the unit-time Lipschitz norm is bounded
by a constant times the initial sup norm. Apply this on the last unit
interval to get (uniform-smoothing). QED.
----------------------------------------------------------------
25. The two spectral gaps
----------------------------------------------------------------
For a tuple with real ordered lift w, write
w-bar = (1/4) sum_i w_i,
r = w - w-bar * 1,
theta = w-bar (mod 2pi).
The relative coordinate lies in the bounded convex ALCOVE
A = { r in R^4 : sum_i r_i = 0, r_1 < r_2 < r_3 < r_4 < r_1 + 2pi }.
This identifies C as (circle) x (alcove). Its invariant law is a product
as well, since J depends only on the relative angles. The center part of
D is (3/32) partial_theta^2 for each tuple.
LEMMA 25.1 (spectral gaps).
(i) The relative-angle diffusion for one tuple has Poincare gap at
least 1/2.
(ii) Consider the subspace V of L^2(mu) consisting of functions that
are invariant under simultaneous rotation of the two tuples AND
change sign under one cyclic relabeling of either tuple.
On V, -D has gap at least 3/4.
(iii) On the subspace of V with both center frequencies zero, -D
has gap at least 1.
PROOF.
Step 1 (one-tuple gap from Brascamp-Lieb). For a mean-zero vector
v in R^4 the Hessian of the negative logarithm of the one-tuple density
satisfies
grad^2(-log J^(4/3))[v, v]
= (1/3) sum_{i < j} csc^2((r_i - r_j)/2) (v_i - v_j)^2
>= (4/3) |v|^2.
The Brascamp-Lieb inequality (with the Kolesnikov-Milman boundary
form) on the convex alcove therefore gives
Var(F) <= (3/4) integral |grad_r F|^2 d mu_r.
The Dirichlet form has factor 3/8, giving the gap
gap of -D (relative one-tuple) >= (3/8) * (4/3) = 1/2.
Use inner convex exhaustions of the alcove and pass to the limit.
Step 2 (centre frequencies). For the two tuples, simultaneous rotation
invariance allows only center frequencies (n, -n) with integer n; the
energy cost is 3n^2 / 16.
Step 3 (cyclic-relabeling signs). A one-step cyclic relabeling sends
theta -> theta + pi/2 and acts on A by a measure-preserving affine
isometry. If a function changes sign under this relabeling, its
relative-angle mean must vanish unless e^(i n pi/2) = -1, i.e. unless
n = 2 (mod 4).
Step 4 (combine). Frequencies NOT congruent to 2 modulo 4 pay at least
1/2 + 1/2 = 1 in relative energy: both tuples contribute the relative
gap 1/2, one of which (the changing-sign side) forces relative mean to
vanish. The remaining frequencies have |n| >= 2 and pay at least 3/4
in center energy (3 * 4 / 16 = 3/4). At (n, -n) = (0, 0) the two
relative means vanish, giving the gap 1.
Step 5 (L^2 decay and no absorbing boundary). These inequalities give
L^2 semigroup decay through the symmetric Dirichlet form
E(F, F) = (3/8) integral |grad F|^2 d mu.
The constant function is in the form domain. Cutoffs approaching the
collision faces have vanishing energy because the density vanishes
with power 4/3 > 1. There is no hidden absorbing boundary. QED.
----------------------------------------------------------------
26. Uniform transmission asymptotics
----------------------------------------------------------------
Let
W_-(p) = e^(lambda p) (T^+ - T^-) / h,
and define the simultaneous-rotation projection
(Pi F)(x, y) = (1/2pi) integral_0^{2pi} F(x, y + phi * 1) d phi.
Equivalently, Pi is projection to center frequencies (0, 0). A one-step
cyclic relabeling switches T^+ and T^- while h is unchanged. Thus W_-
has exactly the symmetries of Lemma 25.1 (ii): simultaneous rotation
invariant, sign-changes under cyclic relabeling.
PROPOSITION 26.1 (uniform transmission asymptotics). There are constants
K >= 0, sigma > 0, and C < infinity such that, uniformly over the open
angle chamber and all sufficiently large p,
||W_-(p)||_infty <= C e^(-3p/4), (odd-decay-a)
||Pi W_-(p)||_infty <= C (1 + p) e^(-p), (odd-decay-b)
T^pm(p, x, y) = e^(-lambda p) (K h(x, y)
+ O(e^(-sigma p))). (rank-one-round)
Moreover,
Lip(W_-(p)) <= C e^(-3p/4), (odd-regularity-a)
||W_-(p + ell) - W_-(p)||_infty
<= C e^(-3p/4) sqrt(ell) (0 <= ell <= 1).
(odd-regularity-b)
The function h is uniformly 1/3-Holder on the closed chamber.
PROOF.
Step 1 (Duhamel). For any transformed solution put F_p = B_p . grad W
(p) + V_p W(p). Lemma 24.1 implies ||F_p||_infty <= C e^(-p) once p is
sufficiently large. Variation of constants gives
W(p) = P_{p - p_1} W(p_1)
+ integral_{p_1}^p P_{p - s} F_s ds. (duhamel)
This follows directly by Ito's formula with D, first stopping on inner
exhaustions: on the interval of integration the solution and forcing
are bounded, and collisions are not hit.
Step 2 (W_- decay from the 3/4 gap). For W_- the forcing respects
simultaneous rotation invariance and the two cyclic sign changes. Apply
the 3/4 gap (Lemma 25.1(ii)) to (duhamel), then apply unit-time
L^2-to-supremum smoothing to all semigroup times >= 1. For shorter
times use the sup bound on F_s. The convolution with e^(-s) is
O(e^(-3p/4)). After projecting by Pi (which commutes with D), the gap
is 1 and the convolution is O((1 + p) e^(-p)). This proves (odd-decay-a)
and (odd-decay-b). It does not require B_p to commute with independent
rotations: the projected forcing is estimated after projection.
Step 3 (rank-one from a K). For either W^+ or W^- separately, D has a
positive gap off constants: the center gap is 3/32, and the relative
gap is at least 1/2. The same argument as Step 2 shows exponential
convergence of W - integral W dmu to zero. Also
partial_p integral W dmu = integral F_p dmu = O(e^(-p)),
so its mean converges exponentially. The first estimate of (odd-decay-a)
shows the two limiting constants (for W^+ and for W^-) agree; call their
common value K. Multiplication by e^(-lambda p) h proves (rank-one-round).
Step 4 (Lipschitz). The unit-interval Lipschitz estimate in Lemma 24.1,
applied to the homogeneous perturbed evolution for W_-, improves its
Lipschitz bound to C e^(-3p/4). This is (odd-regularity-a).
Step 5 (time-Holder). To obtain (odd-regularity-b), note that the same
diffusion has expected angle displacement at most C sqrt(ell) in time
ell <= 1, uniformly in initial point. Here is a bound valid near a
collision. Apply Ito's formula to squared distance from the initial
lift. Each pair in the scalar product with the singular drift
contributes a constant multiple of (v - v_0) cot(v/2) with
v, v_0 in (0, 2pi), which is bounded above uniformly: use v - v_0 <= v
for v <= pi, and v - v_0 >= v - 2pi for v >= pi. The bounded additional
drift and the Brownian quadratic variation then give
E|Z_ell - Z_0|^2 <= C ell, by localization and Gronwall. The Feynman-Kac
representation over time ell, the spatial Lipschitz bound, and the
bounded potential give (odd-regularity-b).
Step 6 (Holder of h). J is a bounded Lipschitz nonnegative function on
the closed chamber; taking a cube root is 1/3-Holder. The assertion for
h follows by taking products. QED.
================================================================
PART V. SEPARATING THE CONFORMAL DATA OF THE TWO ENDS
================================================================
What this Part delivers.
Part IV computed T^pm(p, x, y) with (p, x, y) expressed in the
uniformization of the ENTIRE annulus. But in Part VI we will transfer
data between two ends of an annulus whose inner and outer boundaries
are JORDAN CURVES OF ARBITRARY SHAPE, not round circles. We need the
same asymptotic formula to hold when p is replaced by the modulus
arising from the two independently normalized ends, and when x, y
are replaced by the separately normalized switch angles.
Deliverable (rank-one-after-separation):
T^pm approx K * l_i^(5/4) J(y^0)^(1/3)
* l_o^(-5/4) J(x^0)^(1/3),
error <= C (a/b)^(5/4 + sigma_1),
for some sigma_1 > 0.
Deliverable (rotation-average signed decay):
(1/2pi) integral over rotation of (T^+_phi - T^-_phi) d phi
has absolute value <= C (a/b)^(2 + sigma_2),
for some sigma_2 > 0 (specifically any 0 < sigma_2 < 1/4).
The pair (rank-one-after-separation, rotation-average signed decay)
has exactly the two properties needed in Part VI's transfer operator:
the leading term depends on the two ends SEPARATELY, and the leftover
symmetric piece decays faster than (a/b)^2.
----------------------------------------------------------------
27. Setup and the end normalization
----------------------------------------------------------------
Let
K_i be a closed Jordan region containing 0 in its interior,
U_o be a bounded Jordan domain containing K_i.
The annulus under consideration is U_o \ K_i. Its two boundary
components are partial K_i and partial U_o.
Fix constants c_*, C_* > 0 and assume:
(shape-hypotheses)
K_i subset { |z| <= a }, diam K_i >= c_* a,
{ |z| < b } subset U_o, min_{z in partial U_o} |z| <= C_* b,
b / a larger than a constant depending only on c_*, C_*.
Each boundary carries four distinct marked prime ends and alternating
color prescriptions (as in Definition 15.1 C3).
Normalize the two ends SEPARATELY.
INNER END. Let
w_i : (C-hat \ K_i) -> { |w| > 1 }, w_i(z) = z / l_i + O(1),
as z -> infinity. Here l_i > 0. Equivalently, l_i is the LOGARITHMIC
CAPACITY of K_i.
OUTER END. Let
w_o : U_o -> { |w| < 1 }, w_o(0) = 0, w_o'(0) = 1 / l_o > 0.
Here l_o > 0. Equivalently, l_o is the CONFORMAL RADIUS of U_o at 0.
Write y^0 (for the inner end) and x^0 (for the outer end) for the
angles of the four marked switches in these maps.
Capacity-diameter bounds for continua and the Koebe estimates give:
(end-scales)
l_i \asymp a, l_o \asymp b,
with constants depending only on c_*, C_*.
----------------------------------------------------------------
28. Uniform end-map comparison
----------------------------------------------------------------
LEMMA 28.1 (uniform end-map comparison). Let (p, x, y) be the modulus and
marked angles in a UNIFORMIZATION OF THE WHOLE ANNULUS U_o \ K_i, with
angular direction the same on both boundary components. Put
p' = log(l_o / l_i).
The common rotational normalization of the annulus map can be chosen so
that
|p - p'| <= C a/b,
|x - x^0| + |y - y^0| <= C (1 + log(b/a)) a/b. (comparison)
Distances in the second inequality use compatible cyclic lifts. The
constant is uniform over all Jordan boundaries satisfying the shape-
hypotheses, INCLUDING all placements of the marks.
PROOF.
Step 1 (analytic corrections). Define single-valued analytic corrections
g_i(z) = log( l_i w_i(z) / z ), (branch vanishing at infinity)
g_o(z) = log( l_o w_o(z) / z ), (branch vanishing at zero).
The quotient defining g_o has no zeros in U_o (the "argument" part of
w_o/z is single-valued because U_o is simply connected and 0 is in the
interior), and the apparent singularity at 0 is removable (both
numerator and denominator have simple zeros at 0). The exterior
quotient has zero winding about infinity (the winding number argument
matches the argument of z at infinity), and its logarithm is normalized
at infinity.
Univalent distortion and end-scales give:
(correction-bounds)
|g_o(z)| <= C a / b (|z| <= C_1 a),
|g_i(z)| <= C a / b (|z| >= c_1 b),
for every fixed C_1 < infinity and c_1 > 0, once a/b is small. The
relevant sets in these inequalities are contained in the respective
map domains.
Let psi_i = w_i^(-1) and psi_o = w_o^(-1). The maximum principle applied
to psi_i(w) / w gives |psi_i(w)| <= a |w| for |w| > 1. Koebe's
distortion bound gives, for psi_o(w),
|psi_o(w)| >= l_o |w| / (1 + |w|)^2.
Thus the inverse exterior map sends 1 <= |w_i| <= 2 into |z| <= 2a,
and the inverse interior map sends, for example, 1/2 <= |w_o| < 1 into
|z| >= c b. These observations are the reason arbitrary roughness of
the physical boundaries does not change the constants.
Step 2 (modulus error via single-valuedness of H - A). Consider
H(z) = log(z / l_i) + g_i(z) + g_o(z).
It is analytic with period 2pi i. Its real part is within C a/b of 0 on
the inner boundary and of p' on the outer boundary:
inner: Re H = Re(log(z/l_i) + g_i(z)) + Re(g_o(z)), and on |psi_i(w)|
= |z| <= l_i, log|z/l_i| = 0, and g_i, g_o are both small.
outer: by the same comparison, Re H is p' up to O(a/b), because
|psi_o(w)| = l_o on |w| = 1 and g_o + g_i is small.
Let A(z) be an annulus logarithm, normalized by 0 < Re A < p. Its period
is also 2pi i. The difference H - A is single-valued. In the cylinder
coordinate A, the circular average of the real part of any single-valued
analytic function is constant across the cylinder. At the two boundary
circles the averages of Re(H - A) are O(a/b) and p' - p + O(a/b).
Equality of the averages proves
|p - p'| <= C a/b.
The maximum principle now gives
(real-error)
|Re(A - H)| <= C a/b throughout U_o \ K_i.
Step 3 (angular comparison; real parts alone are not enough). Use
instead the round collars in the single-end coordinates. The relevant
single-valued functions satisfy
A - log w_i = (A - H) + g_o,
A - p - log w_o = (A - H) + (p' - p) + g_i.
Near the inner end, A - log w_i has real part bounded by C a/b by
correction-bounds and real-error, and its real part is EXACTLY zero on
|w_i| = 1 (both A and Re(log w_i) vanish there in the normalization
that puts the boundary values of Re A at 0 and p). Schwarz reflection
across this circle, followed by interior gradient bounds on a smaller
fixed collar, bounds the variation of the IMAGINARY part along the
entire end circle by C a/b.
Near the outer end apply the same argument to A - p - log w_o. Its
real part is zero on |w_o| = 1 and is bounded by C a/b in a fixed inner
collar. This gives the analogous angular bound, initially with possibly
different additive constants at the two ends.
Step 4 (matching the constants). Choose C_1 large and c_1 small, both
fixed. The round annulus C_1 a < |z| < c_1 b is contained in the
physical annulus. Along a radial segment across it, use O(1 + log(b/a))
overlapping disks of fixed relative size. The gradient estimate for
A - H, whose real part obeys real-error, bounds the variation of its
imaginary part on each disk by C a/b.
At either end, distortion in the corresponding single-end chart connects
this radial segment to the fixed collar with a bounded additional chart
distance. The same interior estimate and the collar reflection estimate
apply there. Thus the two additive constants differ by at most
C (1 + log(b/a)) a/b.
Choose the common rotation to fix the inner constant. The comparison of
all marked angles, uniformly even when marks are very close, follows.
This proves (comparison). QED.
Numerical consequence: if rho = a/b, then p - p' = O(rho) and the
angular error is at most C (1 + log(1/rho)) rho. Define
eta = C (1 + log(1/rho)) rho.
This is the same eta used in the next Proposition.
----------------------------------------------------------------
29. Separated-end transmission
----------------------------------------------------------------
PROPOSITION 29.1 (separated-end transmission). There are constants
sigma_1, sigma_2 > 0 and C < infinity, depending only on the fixed
constants in shape-hypotheses, such that for every marked annulus
satisfying shape-hypotheses:
(rank-one-after-separation)
| T^pm - K * l_i^(5/4) J(y^0)^(1/3) * l_o^(-5/4) J(x^0)^(1/3) |
<= C (a/b)^(5/4 + sigma_1).
Here K is the common constant from Proposition 26.1 (rank-one-round).
(rotation-average-after-separation)
| (1/2pi) integral_0^{2pi} ( T^+_phi - T^-_phi ) d phi |
<= C (a/b)^(2 + sigma_2),
where phi rotates the INNER end (with its marks and colors) via
w_{i, phi}(z) = e^(i phi) w_i(e^(-i phi) z), keeping the outer end fixed.
The same bound holds if the outer end is rotated instead.
Both estimates are uniform for Jordan approximants to the prime-end
boundaries of Part III, provided they satisfy shape-hypotheses with the
same fixed constants.
PROOF.
Step 1 (rank-one after separation). Write
rho = a/b, q = log(1/rho).
By end-scales, p' = q + O(1). Set
eta = C (1 + q) rho,
the angular error from Lemma 28.1.
The Holder bound for h (h is 1/3-Holder on the closed chamber from
Proposition 26.1) and the first inequality of Lemma 28.1 give
| e^(-lambda p) h(x, y) - e^(-lambda p') h(x^0, y^0) |
<= C rho^lambda ( rho + eta^(1/3) ).
(Here the rho term is |p - p'|; the eta^(1/3) term is the h modulus of
continuity applied to the angular error.)
Combine with (rank-one-round):
T^pm(p, x, y) = e^(-lambda p) K h(x, y) + O(e^(-sigma p)),
and e^(-lambda p) K h(x, y) is close to
e^(-lambda p') K h(x^0, y^0)
up to error C rho^lambda (rho + eta^(1/3)).
The leading term e^(-lambda p') K h(x^0, y^0) equals
K (l_i / l_o)^lambda J(x^0)^(1/3) J(y^0)^(1/3),
= K l_i^lambda J(y^0)^(1/3) * l_o^(-lambda) J(x^0)^(1/3),
which is exactly the separated form. Choosing sigma_1 < min{sigma, 1/3}
absorbs the (rho + eta^(1/3)) factor times rho^lambda into rho^(lambda
+ sigma_1) after increasing the constant.
Step 2 (signed rotation average; setup). Put
U = T^+ - T^- = e^(-lambda p) h W_-.
Equations (odd-decay-a) and (odd-regularity-a,b) together with the
Holder bound for h show that replacing the actual annular data by the
two single-end data costs at most
(signed-shape-error)
C rho^2 ( rho^(1/2) + eta + eta^(1/3) ).
This bound is uniform over rotation of either physical end. In
particular it is a POWER improvement on rho^2. (For later use we only
need that signed-shape-error <= C rho^(2 + sigma_2') for some fixed
sigma_2' > 0.)
Step 3 (rotation average in the unperturbed data). The capacity l_i is
unchanged by rotation of the inner end, and its normalized map is
w_{i,phi}(z) = e^(i phi) w_i(e^(-i phi) z). Consequently its marked
single-end angles are exactly y^0 + phi * 1. The factor h is unchanged
under rotating either tuple SEPARATELY (h = J(x)^(1/3) J(y)^(1/3),
and J uses differences of angles, which are rotation invariant).
Thus the rotation average of U in the UNPERTURBED single-end data is
e^(-lambda p') h(x^0, y^0) * (Pi W_-)(p', x^0, y^0),
whose absolute value is at most, by (odd-decay-b),
C (1 + q) rho^(lambda + 1)
= C (1 + q) rho^(9/4).
Add the uniform error (signed-shape-error). The total is
C (1 + q) rho^(9/4)
+ C rho^2 ( rho^(1/2) + (1+q) rho + ((1+q) rho)^(1/3) )
<= C rho^(2 + sigma_2)
for any 0 < sigma_2 < 1/4, after increasing the constant and restricting
to sufficiently small rho. The three summands are:
rho^(9/4) * (1+q): polylog times rho^(2 + 1/4), OK for sigma_2 <= 1/4.
rho^(5/2): OK for sigma_2 <= 1/2.
rho^(7/3) * (1+q)^(1/3): OK for sigma_2 <= 1/3.
(1+q) rho^3: OK for sigma_2 <= 1.
The binding constraint is sigma_2 < 1/4.
Rotating the OUTER end gives the same projection, by simultaneous
rotation invariance (Lemma 20.1). QED.
Step 4 (uniformity over Jordan approximants). Every estimate in the
above proof was obtained with constants depending only on the separation
and scale conditions of shape-hypotheses, not on boundary smoothness or
mark separation. It can therefore be applied DIRECTLY to the continuum
function T evaluated on the conformal data of the Jordan cycles
approximating an explored domain. Under admissible convergence,
Proposition 19.1 separately identifies the limit of the discrete
transmission probabilities with that continuum function; no reflection
across a physically nonsimple limiting bank is being asserted.
================================================================
PART VI. UNIFORM TRANSFER OF SIGNED CONNECTION STATES
================================================================
What this Part delivers.
The two Corollaries at the end of Part VI are the whole point:
tr:insertion (fair-insertion message):
|| I_s ||_s <= C s^(-2 - nu), some 0 < nu < 1,
for a difference of inside-state laws induced by a single fair
insertion at a cut of radius s. Exponent -2 - nu STRICTLY beats
-2: this makes the insertion response integrable over the 2D
plane (used in Part VII).
tr:wedge (contraction of antisymmetric products):
|| M u wedge M v ||_pi <= C (s/r)^(-2 lambda - nu'),
some 0 < nu' fixed. The wedge product of two transferred messages
decays strictly faster than (s/r)^(-2 lambda) = (s/r)^(-5/2).
This gives a small error in the double-difference determinant of
Part VII.
Both follow by iterating a single fixed-ratio block estimate:
the block transfer M_{r, Lr} is essentially rank-one with a
signed-colour cancellation on even-isotropic inputs.
Why "uniform over arbitrary signed data"?
In the discrete Voronoi model the outside of a cut is an ARBITRARY
signed measure on connection states. It has no scaling limit. The
block estimate must therefore be uniform in the input message, not
merely in some typical input. That is why we phrase everything as an
operator norm on a Banach space of signed measures.
----------------------------------------------------------------
30. Seams, connection states, and messages
----------------------------------------------------------------
Fix a sufficiently small eta > 0 for the whole Part. A SEAM at radius r
consists of:
- the marked Poisson process in the collar (1 - eta)r < |z| < (1 + eta)r,
- auxiliary randomness used to select a graph cycle.
A local occupancy test G_r (the "screen") ensures that the graph and the
vertex stars through a NARROWER collar are determined by these variables
and that their diameters are smaller than a fixed small fraction of
eta * r. We choose the test and the narrower collar with buffers at
BOTH ends, so that
P(G_r^c) <= C exp(-c r^2). (tr:screen)
The test and the cycle choice are rotation-equivariant; an independent
uniform angular axis is included in the seam variables to break ties.
Concrete realization. Require that every point in a closed intermediate
collar be within a sufficiently small distance (proportional to r) of a
nucleus in the larger collar. This covering-distance condition is
rotation invariant and has the exponential failure bound in
(tr:screen). Following a circle in the site triangulation and deleting
loops gives the required cycle. Buffers and the same covering condition
determine all its vertex stars. A nucleus beyond either extremity of the
collar cannot affect the graph on the other side: its cell would have to
pass through the screened intermediate collar, where all cells are
small. Consequently the graph on each side of Gamma_r (including its
attachments to the cycle) can be read from the seam and the Poisson
variables on that side. CONDITIONAL ON THE SEAM, THE VARIABLES USED ON
THE TWO SIDES ARE INDEPENDENT.
On G_r choose a simple site cycle Gamma_r surrounding the center,
inside the narrower collar. The choice uses the geometry but NOT the
colors.
Connection states on Gamma_r. The colors on Gamma_r determine an even
number of SWITCHES. An INSIDE CONNECTION STATE is a noncrossing pairing
of those switches by interface strands in the disk on the inside. The
pairing, together with the monochromatic runs on the cycle, determines
all boundary connectivity. Closed interface loops carry no information
and are not stored. We allow EVERY formal noncrossing pairing as a
state, whether or not it is realized by a particular inside graph.
Such pairings respect the alternating color convention: the endpoints
of each pair have OPPOSITE PARITY in cyclic order. Gluing a pairing to
annular strands is therefore defined without a realizability restriction.
Write Sigma_r for the seam variables and S_r(Sigma_r) for the (finite)
state set on G_r.
A MESSAGE is a measurable field of signed measures v_{Sigma_r} on
S_r(Sigma_r) such that
v_{Sigma_r}( S_r(Sigma_r) ) = 0, v_{Sigma_r} = 0 on G_r^c.
Its norm:
||v||_r = ess sup over Sigma_r of || v_{Sigma_r} ||_{TV}. (tr:message-norm)
Here total variation is the sum of absolute point masses; a difference
of probability measures has norm at most two. Finite state sets can be
numbered measurably; the bounded fields of zero-mass measures, modulo
seam-law null sets, form a Banach space B_r.
----------------------------------------------------------------
31. Transfer operators
----------------------------------------------------------------
If the collars at radii r < s are disjoint, gluing through the
intervening annulus defines a linear map
M_{r, s} : B_r -> B_s.
More explicitly, conditional on Sigma_s, integrate Sigma_r and the
intervening marked process, impose both seam successes, and push
v_{Sigma_r} through the annular strands. Closed loops have weight one.
For every fixed realization, this push is a map between finite state
spaces and contracts total variation. Its mass remains zero. The
success restrictions are INDICATORS in this integral, not a conditioning
followed by a renormalization. Only the output seam is conditioned on
in the definition of its message field. Intermediate successful seams
therefore give EXACTLY the composition of these operators, with the
corresponding success indicators imposed.
This construction does not assert independence for unrestricted Voronoi
events in neighboring regions; independence is used only for the
underlying marked Poisson variables separated by the specified buffers.
Symmetries. Rotation and global color reversal act on seam variables
and states. A message is ISOTROPIC if fixed by the rotation action
and EVEN if fixed by color reversal. Both symmetries are exact for
M_{r,s}, because the screens and cycle choices have those symmetries.
----------------------------------------------------------------
32. The fixed-ratio block estimate
----------------------------------------------------------------
PROPOSITION 32.1 (fixed-ratio transfer). There are delta > 0 and C <
infinity with the following property. For every sufficiently large
integer n, set L = 2^n. There is a threshold r_0(L) such that, for
r >= r_0(L), the transfer M = M_{r, Lr} satisfies
|| M || <= C L^(-lambda),
(tr:block-size)
M = M_1 + E, rank(M_1) <= 1,
|| E || <= C L^(-lambda - delta),
(tr:block-rank)
|| M on B_r^{even, iso} || <= C L^(-2 - delta).
(tr:block-cancellation)
The constants C, delta do not depend on L or r. Every norm in the
conclusion includes the essential supremum over successful output
seams, INCLUDING THEIR COLORS. The rank-one operator M_1 may depend
on r and L.
Recall lambda = 5/4.
Mechanism of the proof (we do not prove the proposition here; Sections
33-36 do). Six surviving interfaces already cost a power strictly
greater than two (via the six-arm exponent from [V]). Four surviving
interfaces leave just ONE signed connection coordinate (the cap
difference in Lemma 13.1). Summing over the scales at which the
transfer first reduces to exactly four tips gives both the rank-one
approximation and the signed colour-rotation cancellation. Uniformity
over arbitrary signed states comes from the domination Lemma 33.2
(Section 33).
----------------------------------------------------------------
33. An independent region for the limiting argument
----------------------------------------------------------------
Fix L = 2^n for the moment and rescale by r. Thus the mesh tends to
zero as r -> infinity, whereas ALL RADII USED IN THE PRESENT ARGUMENT
lie in a fixed finite range. Choose AUXILIARY STARTING CYCLES near
radii 8r and Lr/8. Their geometry is determined in small collars
separated from both original seams.
From either starting cycle we explore toward the middle of the annulus,
stopping at one of finitely many BARRIERS: color-independent site cycles
approximating specified concentric circles in curve order. Their colors
remain UNREAD until the exploration proceeds past them. A STOPPING
CIRCLE is the limiting circle of the current barrier. All cycles may be
made chord-free on the side approached by the exploration.
We impose auxiliary occupancy tests in this intervening region. Unlike
the actual seam tests, these may require a RELATIVE mesh tending to
zero: use covering distances that tend slowly to zero in rescaled
coordinates. For the fixed finite collection of collars under
consideration, their total failure probability is o(1). Their supports
are separated from the input and output seams.
On success, the geometry around the two starting cycles and throughout
the region between them agrees with the geometry of a FULL-PLANE Poisson
sample, obtained by adjoining independent variables outside that region.
In particular, all these o(1) failure bounds hold UNIFORMLY conditional
on either or both original seam variables.
Local version. An end experiment stopped at a barrier uses only variables
from a buffer just BEFORE its starting cycle through a small fixed-factor
enlargement PAST that barrier, together with the variables between its
starting cycle and its own seam. Its localization tests stop at that
enlargement. If two barriers are separated by a sufficiently large
fixed factor, the SUPPORTS for the two end experiments are DISJOINT.
Extra tests across the middle can then be added when evaluating an
actual annular transmission. DELETING those extra tests from an integral
with bounded weights changes it by o(1), uniformly in the original
seams. It is only AFTER this deletion that the two end integrals are
factored.
Radial order for two end experiments. On a logarithmic scale, from
inside to outside:
input seam
inner auxiliary start (8r)
... inner barriers a_1, a_2, a_3, ...
(unexplored annulus)
... outer barriers ..., b_3, b_2, b_1
outer auxiliary start (Lr/8)
output seam
The two end supports (inner starting region plus buffers up to its last
barrier; outer starting region plus buffers up to its last barrier)
are disjoint in the unexplored annulus. This gap is what permits their
independent integration. The original seam screens remain imposed
throughout.
The uniformity mechanism we use below.
LEMMA 33.2 (domination before conditioning on end shapes). Suppose a
contribution to M_{r, Lr} v is expressed, after exploring from the
auxiliary starting cycles, as an integral of an inner scalar coefficient,
a bounded middle test, and an outer signed state measure.
If the middle test is replaced with absolute error F >= 0, where F
depends only on the independent region just described, then the error
in output norm is at most
C ||v||_r E_core F + o(1) ||v||_r. (tr:domination-bound)
The same bound applies to DISCARDING an event there, by taking F to be
its indicator (0 or 1). The expectation is under the UNWEIGHTED
FULL-PLANE COMPARISON LAW; the bound is uniform over successful output
seams.
PROOF.
Step 1 (total variation contraction). For a fixed realization, pushing
v_{Sigma_r} to any intermediate state set contracts total variation.
When that set has TWO states (the reduction to four tips later), its
zero-mass image has the form alpha (delta_B - delta_W) with
|alpha| <= ||v||_r. Pushing the two-state difference through the other
end gives a signed measure q with ||q||_{TV} <= 2. Hence the pointwise
norm of the replacement error is bounded by 2 ||v||_r F.
The same argument, summing absolute masses BEFORE a two-state reduction,
handles discarded events. No regularity of the dependence of alpha or
q on the seam data is needed.
Step 2 (independence). On the auxiliary localization successes, F is a
function of marked Poisson variables INDEPENDENT of both original seams
and has the stated full-plane comparison law. Integrate the pointwise
bound conditional on the output seam. Auxiliary failures contribute
the final o(1) term, also uniformly, because their supports are off the
original seams. The seam success indicators themselves are NOT deleted.
QED.
The purpose of the auxiliary-cycles construction (rather than starting
the end experiments at the original seams) is precisely to have
measurability of |t_r - T_r| under the independent-region law. See
Section 35.
----------------------------------------------------------------
34. Exploration up to the first four-tip frontier
----------------------------------------------------------------
Step 1 (attach input, then explore). First push the input message from
its original seam to the inner auxiliary starting cycle, using the
marked variables in the intervening attachment region. The subsequent
exploration uses the colored starting cycle and the variables toward
its barrier; it NEVER consults an attached cap state. Dependence on the
input message stays in the pushed signed coefficients. At the outer end
the corresponding attachment operation pushes states from its auxiliary
cycle toward the original output seam.
Step 2 (tracing walls). At the inner auxiliary cycle, reveal its colors
and trace every switch into the annulus. A wall is followed until it
pairs with another traced wall OR until its next color query would read
a vertex on the current barrier. In the second case, STOP BEFORE that
query. The stopped wall ends at a switch edge whose next triangle has
its opposite vertex on the barrier. No barrier colors have been read.
The analogous outer experiment is directed inwards. Increasing the inner
barrier radius by a factor two, or decreasing the outer barrier radius
by a factor two, CONTINUES THE SAME WALL SYSTEM. Tracing all walls in a
previously colored sample shows the outcome does not depend on their
tracing order.
Step 3 (surviving tips). Call the unpaired stopped wall ends SURVIVING
TIPS. Their number is even and decreases as the barrier is advanced.
The FIRST BARRIER with at most four surviving tips is the end's
REDUCTION barrier.
- Reduction to zero or two tips kills every zero-mass message.
- Reduction to exactly four leaves precisely two cap states.
We now verify both assertions and the geometric properties needed for
the annular transmission limit.
LEMMA 34.1 (frontier and arm counts). For either end experiment, outside
an event whose probability tends to zero as the permitted separation of
tips tends to zero, MESH FIRST, the tips at each of finitely many fixed-
scale barriers have distinct limiting positions. The boundary facing the
unqueried barrier is a simple site cycle whose switches are exactly these
tips. In particular:
(i) If there are zero or two tips, the induced map on zero-mass
connection-state measures is zero.
(ii) If there are at least six tips, six disjoint bulk arms cross the
region from the starting cycle to the barrier, after fixed-factor
crops at the two ends.
All probabilistic assertions are made under the UNWEIGHTED EXPLORATION
LAW, before conditioning on a particular history.
PROOF.
Step 1 (colliding tips). Consider two tips approaching the same point
of the barrier. Use a graph disk patch with base on the barrier and a
half-annulus on the exploration side. Extend each stopped wall through
its last triangle to the base, only for this geometric argument; no
barrier color is queried.
The two wall fingers give two proper traversals of the half-annulus,
with fixed margins at its two semicircular ends. Following the sites on
their banks gives THREE disjoint free arms in the sectors on the two
exterior sides and between the fingers. These sites are on the
exploration side, not on the barrier. Short spokes through successive
wall triangles keep each bank in its sector. The base through the tips
prevents an exterior bank from escaping around a finger's end. Cropping
the last mesh steps gives the same count when two stopping triangles
share a barrier vertex.
This is the free-bank-shadow count for two traversals in the boundary
argument of [V, Section 3]. The local boundary three-arm exponent is
strictly greater than one. Covering the barrier by O(s^(-1)) patches of
radius s therefore excludes colliding tips as s -> 0, MESH FIRST. The BK
inequality applied to the free shadows also makes the number of fixed-
scale traversals tight.
Step 2 (frontier boundary is simple). Fill the starting side of the
starting cycle with a disk, using the sphere for the outside experiment.
Attach the queried triangles along the explored bands and add all edges
between revealed vertices. The resulting graph is TWO-CONNECTED:
attaching a triangle along an existing edge preserves two-connectivity,
as does adding an edge between existing vertices. The boundary of its
face containing the unqueried barrier is a SIMPLE CYCLE. The barrier
and its paths toward the unexplored middle remain in that face. In
particular, each stopping triangle is incident with this same face on
its unread side.
Step 3 (color invariant). Invariant: every revealed vertex has a
revealed path OF ITS OWN COLOR to the starting cycle.
Initial: starting-cycle vertices belong to their own run of the
starting cycle. Inductive: a newly queried triangle vertex has the
color of ONE ENDPOINT of the incoming heterochromatic edge, extending
such a path. Thus an added heterochromatic edge between revealed
vertices CANNOT belong to an unstarted wall: such a wall would either
close in the annulus or return to the barrier without visiting the
starting cycle, and either way would separate one bank from that cycle,
contradicting its revealed same-color path. Every heterochromatic
revealed-vertex edge therefore belongs to the traced wall system.
Step 4 (colors along the frontier). To identify the colors along the
frontier, extend the stopped walls to the barrier as above. These
proper arcs, together with the walls that paired back to the starting
cycle, divide the explored collar into regions with CONSISTENT BANK
COLORS. Every revealed vertex is either a starting-cycle vertex or was
queried in a wall triangle. Its color agrees with the relevant bank.
An added edge cannot cross one of these wall arcs, and Step 3 excludes
any untraced switch there. Previously crossed switch edges have
triangles on both sides in the revealed bands. The only switches exposed
to the remaining face are consequently the SURVIVING STOPPING EDGES.
Step 5 (final claims). With zero or two switches there is just one
noncrossing cap pairing: every input state gives that same pairing, so
a zero-mass difference vanishes (claim (i)).
With six or more surviving wall fingers, their bank shadows give six
DISJOINT SITE ARMS in successive sectors between the starting cycle and
the barrier. Fixed-factor crops remove the two ends where different
shadows could otherwise meet. The arbitrary-color six-arm bound (E-iii)
is applicable, so no particular numbering or color order of the six
arms is required (claim (ii)). QED.
----------------------------------------------------------------
35. Admissibility of four-tip frontiers
----------------------------------------------------------------
The next lemma verifies the non-Jordan boundary issue rather than
assuming that scalar Cardy convergence applies to a random annulus.
LEMMA 35.1 (admissibility of four-tip frontiers). At a fixed barrier,
restrict the unweighted exploration law to the event of EXACTLY FOUR
surviving tips. Along subsequences with good geometry as in [V, Section
2], the four monochromatic frontier arcs are tight in curve order. In
every coupled subsequential limit, after constant pauses are erased,
each arc is SIMPLE and meets the stopping circle ONLY AT ITS TWO
PRESCRIBED ENDPOINTS. The four endpoints are distinct.
For two such experiments ending at WELL-SEPARATED barriers, their
residual annulus satisfies the admissibility conditions of Proposition
19.1. If t_r is the discrete THROUGH-TRANSMISSION PROBABILITY conditional
on the complete geometry and the two end histories, and T_r is the
CONTINUUM TRANSMISSION FUNCTION evaluated on its actual conformal data,
then, for FIXED L and the fixed finite list of barriers,
E_core [ 1_{four tips at both ends} * |t_r - T_r| ] = o(1).
(tr:mean-limit)
The expectation concerns UNWEIGHTED HISTORIES, not a uniform conditional
estimate for all possible histories.
PROOF.
Step 1 (tightness). The BK and successive-excursion argument gives
tightness of the four frontier arcs. Repeated annular traversals provide
disjoint free arms; all sites used (including on the auxiliary starting
cycle) have their ORIGINAL fair colors. This is applied first at fixed
radii and then at successively smaller excursion scales. The contact
arguments below are the face-boundary arguments in [V, Section 3]; we
give the details relevant to the present frontier.
Step 2 (no extra barrier visits). Every frontier vertex off the
starting cycle belongs to a revealed wall band. On its revealed side,
it has an OPPOSITE-COLORED NEIGHBOR with a bank path back to the
starting cycle. Consequently, a visit of a frontier arc to the stopping
circle OTHER THAN AT ITS DESIGNATED ENDPOINT, with a positive excursion
on both sides of the visit, gives three free arms in a half-patch: two
along the arc and one along the opposite bank. The same argument
excludes an intermediate return to an endpoint. The boundary three-arm
exponent and a covering of the circle exclude all such visits. A
nonconstant interval contained in the circle is excluded by the one-arm
bound at a countable dense set of circle points. Constant pauses can be
removed from the parametrization. Distinctness of the four tips follows
from Lemma 34.1.
Step 3 (no self-contact away from the circles). Consider a self-return
of one monochromatic frontier arc at a point AWAY from the starting and
stopping circles. Separate the two visits by a fixed positive excursion
scale. In a smaller annulus around the proposed contact the arc supplies
four proper traversals. Its opposite bank paths lie on the REVEALED
SIDE of the frontier cycle; they may all be JOINED THROUGH THE STARTING
CYCLE (this joining need not be monochromatic at the starting cycle: the
working annulus misses that cycle, and within the working annulus every
part of the joining set is genuinely of the opposite color).
At least two complementary sectors have a frontier traversal on their
boundary whose revealed side faces the sector. Each such sector contains
an OPPOSITE-COLORED LONGITUDINAL CROSSING. If the joined neighbor set
did not cross the sector longitudinally, planar duality would provide a
transverse crosscut avoiding it. Start this crosscut from that frontier
traversal and stop at its first subsequent hit of the frontier cycle. In
the disk on the revealed side, the two traversal ends and the two
crosscut ends alternate along the frontier. The crosscut therefore
separates their neighbor spokes, contradicting the joined neighbor set
(which lies on the revealed side and avoids the crosscut).
The argument remains valid when the traversals BACKTRACK: choose the
transverse cuts along intermediate graph cycles, or apply Hex duality
with such a cycle temporarily wired in the sector. Two disjoint cuts
bound a quadrilateral whose side arcs belong to the chosen traversals
and whose interior avoids those whole proper traversals, not merely
the selected side subarcs. Thus the crosscut's first subsequent frontier
hit lies OUTSIDE the whole traversal from which it started, so the
alternating boundary order used above persists under backtracking. The
crosscut is in the quadrilateral's interior whereas the end spokes are
beyond the cuts; they avoid the crosscut even if other portions of the
frontier enter the sector. Cropping the margins leaves two disjoint
opposite-colored arms in different sectors, in addition to the four arms
along the monochromatic arc. This gives SIX disjoint bulk arms. The bulk
exponent strictly greater than two (E-iii) excludes these contacts by a
covering argument.
At a self-contact ON THE STARTING CIRCLE, the four arc traversals alone
suffice: the arbitrary-color four-arm exponent is strictly greater than
one, and the possible contact locations lie on a fixed circle.
Step 4 (countable scales). All exclusions are made for DISTINCT visits
with FIXED positive excursion scales BEFORE those scales decrease.
Countably many scales and finite ball coverings make the assertions
simultaneous. They bound JOINT probabilities with the four-tip event; no
division by the probability of a rare history is involved.
Step 5 (identify the remaining domain). Cut along the barrier, using
the four short incidences from the tip edges through their terminal
triangles. Between consecutive tips, the frontier arc and its
corresponding barrier interval bound a LOBE POLYGON. The four lobes
attach over open intervals to the untouched region on the middle side
of the barrier. Each lobe converges in order to a Jordan lobe, because
its frontier arc is simple and meets the barrier only at its endpoints.
Different lobe boundaries may touch in physical projection; their
interiors remain disjoint and their distinct accesses are retained. With
two well-separated barriers there is a FIXED OPEN ANNULAR REGION between
the ends.
The adjoining of lobes still gives an annulus, even when boundaries
touch. On either side of the open annular region, start with the closed
disk bounded by the corresponding stopping circle (using the sphere for
the outer side). The complementary set on that side is obtained by
removing the lobe interiors and their open attachment intervals. Each
lobe's free frontier arc remains in this complement: it cannot enter
another lobe's interior (Jordan interiors disjoint) and meets the
stopping circle only at its own endpoints, never on another open
attachment.
Connect a point of the remaining closed set to the preserved
starting-side region by a path in the closed disk whose interior avoids
the stopping circle. Whenever the path enters a lobe, replace the portion
from FIRST ENTRY to LAST EXIT by the intervening subarc of that lobe's
FREE FRONTIER. The entry and exit lie on this free arc. The replacement
avoids every lobe interior and every open attachment, so it cannot undo
previous replacements. There are only four lobes; finitely many
replacements put the path in the complementary set. Thus each side has
a connected complement. The two such nondegenerate continua are
separated by the open annular region and exhaust the complement of the
residual domain. The domain is therefore DOUBLY CONNECTED, with its two
boundary components PHYSICALLY APART.
Lemma 16.1 now gives uniform convergence of the closed-annulus
parametrizations with their prime-end order. The boundary traversal
consists of finitely many INDIVIDUALLY SIMPLE arcs. These are exactly
the hypotheses of Proposition 19.1 (transmission limit).
Step 6 (finish). Extract from any sequence of scales a further sequence
on which the localized full-plane geometries are good in the sense of
[V, Section 2]. Conditional on those geometries, couple convergent
subsequences of the frontier histories just described. All unqueried
colors in the residual face remain independent and fair. Apply
Proposition 19.1 path-by-path to the converging graph annuli. The
continuum function on the actual approximating conformal data also
converges by continuity. Bounded convergence proves (tr:mean-limit)
along the further sequence and hence along the original sequence. Unused
colors on the removed sides of the frontiers may be revealed or
integrated in their own end variables; they do not affect this argument.
QED.
The probabilistic conclusions of Lemmas 34.1 and 35.1 all concern the
INDEPENDENT REGION. Lemma 33.2 therefore makes their discarded
probabilities and the error in (tr:mean-limit) UNIFORM FOR THE OPERATOR
NORM, even when a message depends arbitrarily on the seam and on its
attachment information. This is the reason the explorations start at
auxiliary cycles rather than at the original seams.
----------------------------------------------------------------
36. The first-reduction decomposition
----------------------------------------------------------------
Index the inner barriers by radii
a_j = 8 * 2^j * r, j = 1, 2, ...,
and the outer barriers by radii
b_k = 2^(-k) * L r / 8, k = 1, 2, ....
Each list is finite, ending before the opposite starting region. Fix a
large integer d so that
j + k <= n - d (tr:used-pairs)
ensures disjoint buffered end supports and sufficient separation for
Proposition 29.1 (separated-end transmission).
A USED PAIR is a pair (j, k) satisfying (tr:used-pairs) at which the
two experiments FIRST REDUCE TO EXACTLY FOUR TIPS. (A first reduction
to fewer than four contributes zero by Lemma 34.1.)
Choose tau > 0 smaller than the excess of the mesh-first six-arm
exponent over 2. Let rho_j^in and rho_k^out be the SUBPROBABILITY
LAWS of the corresponding first four-tip histories, including local
screens and the variables attaching each history to its original seam.
- The INNER law integrates the input seam.
- All OUTER laws and outer integrals below are conditional on the
fixed output seam Sigma_{Lr}; they define fields over that seam,
not averages of it.
We first bound the masses of these unweighted history laws; pointwise
bounded message factors will then preserve the bounds up to constants.
LEMMA 36.1 (first-reduction tails). For fixed L, the end-history
subprobability masses satisfy, in the mesh-first limit,
|| rho_j^in ||_{TV} * ess sup_{Sigma_{Lr}} || rho_k^out(.|Sigma_{Lr}) ||_{TV}
<= C 2^(-(2+tau) j) 2^(-(2+tau) k) + o(1) (tr:history-weight)
for each used pair (j, k). Consequently, after inserting arbitrary
middle tests of modulus at most one, the same bound up to a constant
holds for the corresponding operator contribution on inputs of norm
at most one. The part of the transfer NOT represented by used pairs
has norm at most
C (n+1)^2 L^(-2-tau) + o(1). (tr:unused)
Both estimates are UNIFORM for input messages of norm at most one and
for successful output seams.
PROOF.
Step 1 (mass bound at used pairs). Before the first reduction at level
j, there are AT LEAST SIX TIPS at the preceding level. Lemma 34.1 then
requires six arms from the starting region to that level. The mesh-first
six-arm bound (E-iii) gives
C * 2^(-(2+tau) j),
with a changed constant for the first few levels. The same statement
holds at the OUTER end, uniformly in its original seam: the necessary
six-arm event is read in the independent region, while attachment
variables and success indicators have total conditional mass at most
one. Multiplying the two separate mass bounds proves (tr:history-weight).
For used pairs the physical end experiments also have DISJOINT BUFFERED
SUPPORTS, so their joint law is the product used in the operator
contribution. Arbitrary input weights are bounded pointwise by their
total variation (Lemma 33.2); the number of switches on the input seam
does not enter the estimate.
Step 2 (unused contribution). Run the inner exploration to its LONGEST
allowed level. If it never reduces, it has paid a six-arm probability
over n - O(1) dyadic levels. Otherwise let j be its first reduction
level. If the reduction has fewer than four tips, its contribution is
zero. In the remaining case, run the outer experiment as far as
(tr:used-pairs) allows, possibly through no additional level. If it
does not reduce in time, the two experiments together have paid
six-arm probabilities over at least n - O(1) levels. Their buffered
supports remain disjoint. Summing the resulting bounds over the at
most O((n+1)^2) possible terminal levels proves (tr:unused).
The finitely many auxiliary localization failures and near-coincidences
of tips cost o(1) for fixed L, uniformly by Lemma 33.2. QED.
At a four-tip inner frontier, let B and W denote respectively the
black-join and white-join cap states (Lemma 13.1). The pushed input
message is
alpha_j(v; H_i) * (delta_B - delta_W), |alpha_j(v; H_i)| <= ||v||_r,
(tr:inner-coefficient)
where H_i includes the inner history and its attachment to the input
seam. The coefficient alpha_j is LINEAR in v.
Similarly, pushing (delta_B - delta_W) from an outer four-tip frontier
to the output seam gives a signed measure q_k(H_o) satisfying
|| q_k(H_o) ||_{TV} <= 2. (tr:outer-coefficient)
Lemma 13.1 (cap identity) says the residual annulus sends the first cap
difference to the second multiplied by its through-transmission
probability. Thus the used-pair contribution has the form
integral alpha_j(v; H_i)
* E_mid[ 1_{G_aux} * t_r | H_i, H_o ]
* q_k(H_o)
d rho_j^in(H_i) d rho_k^out(H_o), (tr:integral)
where G_aux denotes the additional middle screens and E_mid integrates
the middle Poisson geometry conditional on the two end histories. The
probability t_r already integrates the fresh colors in the remaining
face, conditional on all geometry. The end laws are a PRODUCT because
their buffered supports are disjoint. The conditional factor in square
brackets still COUPLES the ends and must not yet be factored.
Replace t_r by T_r. On the auxiliary localization successes, the
residual frontiers and both t_r and T_r depend only on the auxiliary-
cycle explorations and the intervening independent geometry. Attachments
to the original seams enter only the signed coefficients. The error
|t_r - T_r| thus has exactly the independent-region measurability
required by Lemma 33.2.
By Lemmas 35.1 (tr:mean-limit) and 33.2, replace t_r by T_r in
(tr:integral) with o(1) ||v||_r error in output norm. For fixed n, this
error may be summed over all used pairs. The frontiers themselves are
still the ACTUAL GRAPH CYCLES. Their single-end conformal maps and
capacities can therefore be used directly in Proposition 29.1, without
asserting smoothness at contacts of a limiting boundary.
----------------------------------------------------------------
37. Rank-one factorization and signed cancellation
----------------------------------------------------------------
We now complete the proof of Proposition 32.1.
Step 1 (geometry of the four-tip frontiers). An inner frontier at level
j surrounds the center, stays inside radius comparable to a_j, and
reaches that radius at its tips. Its logarithmic capacity l_i is
therefore comparable to a_j. An outer frontier at level k surrounds a
disk of radius comparable to b_k and has a point at distance comparable
to b_k; its conformal radius l_o is comparable to b_k. Write y^0 and x^0
for the corresponding SINGLE-END switch angles.
By Proposition 29.1 (rank-one-after-separation),
T_r(H_i, H_o) = K U(H_i) V(H_o)
+ O((a_j / b_k)^(lambda + sigma_1)), (tr:separated-formula)
where
U(H_i) = l_i^lambda J(y^0)^(1/3),
V(H_o) = l_o^(-lambda) J(x^0)^(1/3).
These factors depend on THEIR OWN end histories only, and
| U(H_i) V(H_o) | <= C L^(-lambda) 2^(lambda (j + k)).
(tr:factor-size)
Step 2 (rank-one operator). For the SEPARATED leading term, delete the
extra middle screens. This changes the integral by o(1) ||v||_r for
fixed L: the screens use independent interior variables and fail with
probability o(1), while all weights at these fixed scales are bounded.
Keep the original seam screens and the local end screens. The remaining
end experiments now use disjoint Poisson supports for EVERY used pair.
Define the scalar functionals and output messages
u_j(v) = integral alpha_j(v; H_i) U(H_i) d rho_j^in(H_i),
v_k = integral V(H_o) q_k(H_o) d rho_k^out(H_o).
The leading contribution of that pair is exactly K u_j(v) v_k. The same
definitions make sense separately for every level in each finite list,
WHETHER OR NOT a pair of levels has disjoint supports. For such additional
pairs we use the product of the TWO SEPARATE integrals, not a claim about
simultaneous experiments.
Set
M_1 v = K ( sum_j u_j(v) ) ( sum_k v_k ). (tr:rank-one-definition)
This operator has rank at most one. By the separate six-arm bounds
(tr:history-weight) and (tr:factor-size), its norm is at most
C L^(-lambda) sum_{j, k} 2^(-(2 + tau - lambda)(j + k))
<= C L^(-lambda). (tr:rank-one-size)
The extra product terms introduced by allowing j + k > n - d have norm
at most
C L^(-lambda) sum_{j + k > n - d} 2^(-(2 + tau - lambda)(j + k))
<= C (n+1)^2 L^(-2 - tau). (tr:artificial-pairs)
This numerical estimate requires no independence for overlapping
actual experiments.
Step 3 (rank error). Decrease sigma_1 if necessary so that
lambda + sigma_1 < 2 + tau. The errors in (tr:separated-formula),
summed using Lemma 36.1, have norm at most
C L^(-lambda - sigma_1)
sum_{j + k <= n - d} 2^(-(2 + tau - lambda - sigma_1)(j + k))
+ o(1)
<= C L^(-lambda - sigma_1) + o(1). (tr:rank-error-sum)
Together with (tr:unused) and (tr:artificial-pairs), this proves
(tr:block-size) and (tr:block-rank) of Proposition 32.1, with a smaller
positive exponent saving delta < sigma_1. All errors are measured in
OUTPUT TOTAL VARIATION, not merely after testing a fixed continuum
event.
Step 4 (even-isotropic input). It remains to prove (tr:block-cancellation).
Suppose v is even and isotropic. After the continuum replacement, delete
the extra middle screens as in Step 2. Only T_r then depends on both
ends; all other factors belong to their separate, independently
integrated end laws.
Rotate all inner end data, including the seam and the revealed history.
By equivariance of v, the coefficient alpha_j(v; H_i) in (tr:inner-
coefficient) is UNCHANGED:
alpha_j(v; rot_theta H_i) = alpha_j(v; H_i). (tr:coefficient-rotation)
Reverse all colors of the inner data instead. The message is still
unchanged as a geometrically transformed signed measure, but the
reference vector (delta_B - delta_W) changes sign. Thus
alpha_j(v; flip H_i) = - alpha_j(v; H_i). (tr:coefficient-flip)
The inner end law is rotation and color-reversal invariant. The outer
measure factor is unaffected by either transformation of the inner
data. Labels for the four tips may be chosen independently of colors
using the auxiliary angular axis, and transported under rotations.
Step 5 (flip and rotation averages). For each fixed outer history H_o,
color reversal of the inner data gives the identity
integral alpha_j(v; H_i) T_r(H_i, H_o) d rho_j^in(H_i)
= (1/2) integral alpha_j(v; H_i)
[ T_r(H_i, H_o) - T_r(flip H_i, H_o) ]
d rho_j^in(H_i). (tr:flip-integral)
Now average rotations of the inner data in the right side. Since alpha_j
and the inner law are rotation invariant, its contribution is one half
of the integral against the rotation average of T^+ - T^-, with its
appropriate inner prescriptions. In particular we MUST RETAIN THE SIGN;
replacing this difference by an unsigned four-arm event would lose the
required exponent.
By Proposition 29.1 (rotation-average-after-separation),
| rotation average of (T^+ - T^-) | <= C (a_j / b_k)^(2 + sigma_2),
(rotation-average-V)
uniformly over the outer end. The local end experiments and screens can
be chosen rotation and color-reversal invariant.
Decrease sigma_2 so that 0 < sigma_2 < tau. The used-pair contributions
are bounded in total variation by
C L^(-2 - sigma_2)
sum_{j + k <= n - d} 2^(-(tau - sigma_2)(j + k)) + o(1)
<= C L^(-2 - sigma_2) + o(1). (tr:signed-sum)
Adding the unused contribution (tr:unused) proves the desired power
strictly greater than two in (tr:block-cancellation), with delta < sigma_2.
Step 6 (final numeric choice). Choose delta > 0 smaller than all the
positive savings just obtained (delta < min{sigma_1, sigma_2, tau}),
decreasing it to absorb the polynomial factors in n. For each fixed
sufficiently large n, choose r_0(2^n) so that every o(1) term is
smaller than the required fixed power of L. This proves (tr:block-size)
through (tr:block-cancellation).
NOTICE THE ORDER: the ratio L is chosen first, and the scale threshold
r_0(L) AFTERWARDS. No rate of convergence in Cardy's formula, in the
good-sequence extraction, or near a nonsimple limiting bank is used.
----------------------------------------------------------------
38. Iteration and the fair-insertion message
----------------------------------------------------------------
The block estimates have two consequences. The first uses the signed
symmetry estimate to control a Poisson insertion. The second uses the
rank-one approximation to compare two responses at different distant
boundaries.
COROLLARY 38.1 (fair-insertion message). There are 0 < nu < 1 and
C < infinity such that the following holds. Insert a nucleus at the
center of a cut, with an independent fair color. At a seam of radius s,
take the DIFFERENCE of the inside state laws
"with this insertion" MINUS "without this insertion",
conditional on the seam and with its screen success imposed. Its message
I_s satisfies
|| I_s ||_s <= C s^(-2 - nu) (tr:insertion-bound)
for all sufficiently large s. The estimate remains valid in the
presence of fixed data outside the screened collar that do not enter its
inside. The seam tests and the selected cycle are those of the base
process, WITHOUT the insertion.
PROOF.
Step 1 (setup). Choose a ratio L from Proposition 32.1 so large that the
constant per block in (tr:block-cancellation) is absorbed by a smaller
power saving. Choose s_* > r_0(L) sufficiently large. For given s, use
radii
r_j = L^j r_0, 0 <= j <= N, r_N = s,
s_* <= r_0 < L s_*.
All screens use the BASE process. At a successful seam, the inserted
point CANNOT REACH OR ALTER its cycle, by localization. The two inside
state laws consequently have the same state space and each has mass one.
Their difference is zero-mass, norm at most two, and is EVEN and
ISOTROPIC.
Step 2 (if all screens succeed). Iterate (tr:block-cancellation):
|| I_s ||_s <= C (s / r_0)^(-2 - nu), some 0 < nu < 1.
Since r_0 stays in a fixed compact range of positive scales, this gives
(tr:insertion-bound) for that contribution.
Step 3 (failed screens; partition identity). For the remaining
contribution, group configurations by the LARGEST FAILED RADIUS
v = r_j < s. The exact partition, with the final success kept in every
term, is
G_{r_N} = prod_{i=0}^N 1_{G_{r_i}}
+ sum_{j=0}^{N-1} 1_{G_{r_j}^c}
* prod_{i=j+1}^N 1_{G_{r_i}}.
Step 4 (reset at next successful cut). Form a fresh difference of inside
state laws at the NEXT cut Lv = r_{j+1}, multiplying BOTH laws by the
same base-process indicator 1_{G_v^c}. Earlier screen outcomes are left
UNRESTRICTED.
This weighted difference still has ZERO MASS for each successful seam at
Lv: for each common base configuration, the two state laws have equal
total mass. It is also even and isotropic. The collar at v is DISJOINT
from that at Lv, so conditional on the latter seam its norm is at most
2 * P(G_v^c | Sigma_{Lv}) <= C exp(-c v^2). (tr:reset)
Success at Lv ensures that BOTH configurations use the SAME cut there,
independently of what happened at smaller radii.
Step 5 (transfer the reset message). Transfer this reset message through
the successful cuts ABOVE Lv. Those success restrictions ensure that v
IS the largest failed radius; no restriction on earlier failures was
needed. The sum of these contributions is bounded by
C * sum_{v in {r_0, ..., r_{N-1}}} exp(-c v^2) * (s / (L v))^(-2 - nu)
<= C s^(-2 - nu).
The final inequality follows by summing the exponentially decaying
weights times v^(2 + nu) over the geometric sequence. This proves the
claim. Outside data beyond the final screened collar have no effect on
the inside law or on the argument (stability). QED.
----------------------------------------------------------------
39. Contraction of antisymmetric products
----------------------------------------------------------------
For a message space B_t, define the PROJECTIVE NORM on its algebraic
tensor square:
|| z ||_pi = inf { sum_{i=1}^m || u_i ||_t || v_i ||_t :
z = sum_{i=1}^m u_i (x) v_i, m < infinity }.
We may complete in this norm. Bounded linear functionals l, l' on B_t
satisfy
| (l (x) l')(z) | <= || l || || l' || || z ||_pi.
In particular, evaluating u (x) v - v (x) u gives the DETERMINANT
l(u) l'(v) - l(v) l'(u).
COROLLARY 39.1 (contraction of antisymmetric products). There are
nu' > 0, s_0 < infinity, and a fixed ratio L > 1 such that, for r >= s_0
and s/r = L^N, the transfer
M = M_{s/L, s} o ... o M_{r, Lr}
with ALL intermediate seam successes imposed satisfies
|| M u (wedge) M v ||_pi
<= C (s / r)^(-2 lambda - nu') || u ||_r || v ||_r
(tr:wedge-bound)
for any two zero-mass input messages u, v in B_r. Here
u (wedge) v = u (x) v - v (x) u,
and || . ||_pi is the projective tensor norm. Fixed-factor unused
intervals at the two ends change only the constant. No assertion of
this contraction through a FAILED intermediate cut is made.
PROOF.
Step 1 (one block). Write M = M_1 + E as in (tr:block-rank). Since M_1
has rank at most one,
M_1 (x) M_1 vanishes on antisymmetric tensors.
On the antisymmetric subspace,
|| M (x) M ||
<= 2 || M_1 || || E || + || E ||^2
<= C L^(-2 lambda - delta)
(from the block estimates: ||M_1|| <= C L^(-lambda),
||E|| <= C L^(-lambda - delta)).
Step 2 (choose L). Choose L large enough to ABSORB C into a smaller
exponent saving nu' > 0, simultaneously with its choice in Corollary
38.1.
Step 3 (iterate). Iterate the block estimate and use the elementary
bound || u (wedge) v ||_pi <= 2 ||u||_r ||v||_r to get
(tr:wedge-bound).
The argument holds first for finite tensor sums and then for their
projective-norm completion. Ordinary Markov pushes contract total
variation, so fixed-factor OMITTED end intervals affect only the
constant after rounding to the ratio-L grid. QED.
================================================================
PART VII. THE INTENSITY DERIVATIVE AND THE PIVOTAL AMPLITUDE
================================================================
What this Part delivers. The amplitude theorem (Theorem 2 of Part I).
Part VI supplied two estimates with different roles:
Corollary 38.1 makes the effect of a distant fair insertion
integrable, even after division by the central pivotal probability.
Corollary 39.1 makes the normalized effect of a nearby insertion
almost independent of the size of the rectangle.
Together they give a summable error for the logarithmic derivative
of f. Integrating that derivative pins down f to amplitude precision.
Throughout this Part the Poisson intensity is one, unless otherwise
specified, and lambda = 5/4.
----------------------------------------------------------------
40. Differentiating the intensity
----------------------------------------------------------------
Let X_R(eta) be the indicator of a black left-right crossing of R Q_0
for a colored point configuration eta, with the closed-cell convention
of Theorem 2.
If 0^b and 0^w denote an inserted black and white point at the origin,
put
H_R(eta) = X_R(eta U {0^b}) - X_R(eta U {0^w}).
Thus H_R in {0, 1}, and f(R) = E H_R(eta) for the unit-intensity
fairly colored process. For y != 0, write y^f for an independently
fairly colored inserted point and define
K_R(y) = E[ H_R(eta U {y^f}) - H_R(eta) ]. (cl:kernel)
The expectation includes the extra color. Values at y = 0 and at null
nongeneric insertion locations are immaterial to the integrals below.
In particular, K_R is an insertion difference of a COLOR-pivotal
probability, not a probability of Poisson pivotality.
LEMMA 40.1 (intensity differentiation). The function f is continuously
differentiable on (0, infinity). For every R > 0, K_R is absolutely
integrable and
(R / 2) f'(R) = integral over R^2 of K_R(y) dy. (cl:derivative-eq)
PROOF.
Step 1 (scaling). Let eta_t have intensity t and fair colors. Euclidean
scaling gives
G_R(t) := E H_R(eta_t) = f( sqrt(t) R ). (cl:scaling-intensity)
We justify differentiation of the full-plane expectation before using
this identity. All estimates in this step are uniform for t in a
compact subinterval of (0, infinity).
Step 2 (finite-volume reduction). For n >= n_0(R), let H_{R,n} be the
same test computed from the points in B(0, 2^n), together with the
central insertion. A locality screen in a collar well inside B(0, 2^n)
determines every cell meeting R Q_0, including its intersections with
the boundary. Consequently
H_{R,n} = H_R outside an event of probability <= C_R exp(-c_R 4^n),
H_{R,n+1} = H_{R,n} on that same screen.
Finite-volume Poisson differentiation follows by differentiating the
absolutely convergent Poisson series (see also [LP, Theorem 19.1]).
Equivalently, for a bounded function F of the process in a bounded set D,
(d/dt) E F(eta_t cap D)
= E[ F(eta_t cap D) ( N_t(D)/t - |D| ) ]. (cl:poisson-score)
Here N_t(D) is the number of Poisson points in D.
Step 3 (summability of the derivative). Apply (cl:poisson-score) to
F = H_{R,n+1} - H_{R,n} on D = B(0, 2^{n+1}). The score in parentheses
has second moment |D|/t. By Cauchy-Schwarz, the derivative of
E(H_{R,n+1} - H_{R,n}) is bounded by
C_R 2^n exp(-c_R 4^n).
The differences of the expectations and of their derivatives are
uniformly summable. Hence G_R is C^1 and its derivative is the limit
of finite-volume derivatives.
Step 4 (add-one version of finite-volume differentiation).
(d/dt) E H_{R,n}(eta_t)
= integral_{B(0, 2^n)}
E[ H_{R,n}(eta_t U {y^f}) - H_{R,n}(eta_t) ] dy.
For fixed y, the integrand converges to its full-plane counterpart.
For |y| >= C R + C, a screen surrounding R Q_0 and the origin, at
distance comparable to |y| and strictly short of y, prevents the
insertion from changing any cell used by the test. Its failure
probability is at most C_R exp(-c_R |y|^2). The screen can be taken
inside B(0, 2^n) whenever y in B(0, 2^n) and n sufficiently large; thus
this is also a DOMINATING BOUND for the finite-volume integrands. On
a fixed bounded set, the trivial bound 2 suffices.
Dominated convergence proves absolute integrability and identifies
G_R'(1) with the integral in (cl:derivative-eq). Equation
(cl:scaling-intensity) proves the asserted derivative formula and,
locally at each R, the C^1 assertion for f. QED.
----------------------------------------------------------------
41. An integrable tail after normalization
----------------------------------------------------------------
The exponential localization in Lemma 40.1 is a FINITE-SCALE statement.
We need a bound UNIFORM as R grows, with the small factor f(R) retained.
This is where the signed insertion estimate is essential.
LEMMA 41.1 (normalized insertion tail). There are constants xi > 0,
m_0 < infinity, and C < infinity such that for m_0 <= m <= R/100,
integral_{|y| > m} | K_R(y) | dy
<= C f(R) ( m^(-xi) + R^(-xi) ). (cl:tail-eq)
PROOF.
Use the exponent 0 < nu < 1 from Corollary 38.1 and the one-arm exponent
beta > 0 from the critical inputs. All screens and arm probes below have
fixed-factor buffers. INDEPENDENT PROBES means probes determined by
DISJOINT sets of Poisson coordinates, not unrestricted Voronoi events
in disjoint planar regions.
Step 1 (mixed-difference necessary arms). The integrand in (cl:kernel) is
a MIXED DIFFERENCE: one difference changes the central color, and the
other inserts the point at y. If this mixed difference is nonzero, then
changing the central color changes a crossing for at least one setting of
the insertion, and changing the insertion changes a crossing for at
least one setting of the central color. A screened probe separated from
both changes reads the same base colors in all four configurations.
Thus the necessary-arm implications for a local change (INPUT D) apply
to these common colors. They apply even when the two crossing changes
occur in DIFFERENT configurations.
Step 2 (conditional version). Every formal noncrossing pairing can be
drawn as disjoint wall arcs in a topological disk. Coloring its
complementary regions alternately realizes the prescribed boundary runs
and the pairing. Thus two formal cap states give two planar fillings
agreeing on every exterior probe; the deterministic necessary-arm
argument does not require Poisson geometry INSIDE the replaced disk.
Fix a successful cut about y, its seam, and all exterior variables. The
exterior test is now a function of the inside connection state.
- If a necessary central or enclosing four-arm event does NOT occur,
then the central black-minus-white test is zero for EVERY such state.
- If a necessary side or corner arm event does not occur, then each
crossing test with the central color FIXED is CONSTANT across inside
states.
These are the same planar necessary-arm implications, applied to the two
possible cap states on the cut. The insertion message has ZERO MASS, so
it annihilates the latter constants. Its conditional answer is therefore
bounded by its total variation norm times the necessary probe indicators.
Those indicators are measurable OUTSIDE the cut. Disjoint buffered
supports then allow their probabilities to be MULTIPLIED.
Step 3 (screened arm probes). Use screened arm probes with failures
allowed, whose probabilities are still bounded by a constant times their
intended arm bounds (INPUT C). Group failures by the LARGEST failed
radius v and retain the arm test starting a fixed factor beyond v,
independently of the failure. For a four-arm probe starting at r:
exp(-c v^2) A(C_0 v, R)
<= C A(r, R) (v/r)^g exp(-c v^2) (C_0 v < R),
where g is the polynomial exponent in (A1).
For a side or corner probe, the corresponding endpoint losses are
bounded by C (v/d)^2 or C (v/u)^beta respectively. If a failed screen
crosses the junction between side and corner probes, the lost factors
are therefore only POLYNOMIAL. If the retained test has no remaining
radial range, the same comparison follows from the polynomial lower
bound or directly from the displayed power factor for a side or corner
probe. Summing over geometric v preserves the indicated arm cost as long
as the retained probes and the failure event have DISJOINT BUFFERED
SUPPORTS.
A failure large enough to reach the CENTRAL probe requires a separate
argument below: that probe is then DROPPED, not treated as independent.
An unweighted exponentially small error alone would not suffice when R
is arbitrarily large.
Step 4 (insertions near the center, t = |y|, m < t <= R/100). First
fix a small constant c_1 > 0. Then choose a > 0 sufficiently small
relative to c_1. Choose:
- a cut about y of radius a t;
- a central four-arm probe from b_* to a t;
- a four-arm probe surrounding BOTH insertions from a^(-1) t to aR.
Short or overlapping endpoint ranges are omitted at constant cost. The
buffers of these three constructions can be chosen disjoint.
On cut success, Corollary 38.1 gives a bound C t^(-2 - nu) for the
insertion message. The central color difference requires the FIRST
four-arm probe; transmission of a change from the region containing both
insertions to the crossing requires the SECOND. Thus the SUCCESS
contribution is at most
C t^(-2 - nu) A(b_*, t) A(t, R).
For failures, let v be the largest failed radius about y, with the
initial radius comparable to t.
- If v < c_1 t, the central probe and an enclosing probe retain
DISJOINT SUPPORTS after fixed buffer adjustments. The failed-screen
estimate bounds their total contribution by
C exp(-c t^2) A(b_*, t) A(t, R).
- If v >= c_1 t, DROP the central probe. Retain instead a screened
four-arm probe centered at the origin, of radius C_1 (v + t) to aR,
enclosing both modifications and with its support separated from
the failed screen. The same necessary-arm implication supplies this
probe. When its radial range is nonempty:
exp(-c v^2) A(C_1 (v + t), aR)
<= C A(b_*, R) (v + t)^g exp(-c v^2).
Since t <= c_1^(-1) v, summing over geometric v gives at most
C A(b_*, R) exp(-c' t^2).
If the retained radial range is EMPTY, then R <= C (v + t) <= C' v;
the absolute failure bound and A(b_*, R) >= c R^(-g) give the same
conclusion.
Combining success and both failure cases:
| K_R(y) | <= C t^(-2 - nu) A(b_*, t) A(t, R)
+ C A(b_*, R) exp(-c t^2)
<= C f(R) t^(-2 - nu). (cl:bulk-kernel)
The last line uses quasimultiplicativity and (calibration-comparison).
Consequently (lateral area element in polar coords: 2 pi t dt):
integral_{m < |y| <= R/100} | K_R(y) | dy
<= C f(R) integral_m^inf t^(-1 - nu) dt
<= C f(R) m^(-nu). (cl:bulk-integral)
Step 5 (insertions near a side or corner, R/100 < |y| <= 10R, including
points outside the rectangle). Fix a sufficiently large constant d_*,
then a sufficiently small rho > 0. If C_R is the set of the four corners
of R Q_0, set
d(y) = min{ rho R, max{ d_*, dist(y, boundary(R Q_0)) } },
u(y) = min{ rho R, max{ d_*, dist(y, C_R) } }.
Take R large enough that d_* < rho R. Always
d_* <= d(y) <= u(y) <= rho R.
Write d = d(y) and u = u(y). Constants in the estimates may depend on
d_* and rho.
There is a CENTRAL four-arm probe up to radius cR, where c is fixed and
much smaller than 1/100, disjoint from the other regions used below. Its
cost is at most C A(b_*, R).
- Interior point y: use the insertion estimate on a cut of radius a
small fixed fraction of d. At the bottom cutoff a constant bound
suffices and is bounded by C d^(-2 - nu) because d_* is fixed.
- Exterior point with distance above the bottom cutoff: successful
screening at this scale leaves the quad unchanged. Only screen
failures contribute, with a bound exp(-c d^2), no larger than
C d^(-2 - nu).
If u/d is larger than a sufficiently large constant, the closest part
of the boundary is a STRAIGHT SIDE. A crossing change then requires
three alternating FREE arms in a half-plane probe centered at the
projection onto that side, from radius C d to radius c u. Its cost is
at most C (d/u)^2. If the projection is at a CORNER, the two distances
are comparable and this probe is unnecessary. When u is uncapped and a
sufficiently small fraction of R, retain a ONE-ARM probe centered at the
nearest corner, from radius C u to radius c R. Its cost is at most
C (u/R)^beta. For u CAPPED above, this factor is a fixed positive
constant, so the corresponding probe can be omitted. These are exactly
the local-change arm implications for the unwired, closed-cell crossing
convention. They also cover a modified cell meeting a side or a corner.
The insertion cut, the side probe, the corner probe, and the central
probe can be given DISJOINT BUFFERED SUPPORTS. To arrange this: first
fix the central radius c R, then choose the distance cap rho R much
smaller. A corner probe is used only in a corner neighborhood, far from
the center. Fixed multiples are lost between consecutive probes, which
changes only constants.
To include failures, choose a sufficiently small fixed c_2 > 0.
- For a largest failed radius v < c_2 R about y, the central probe
remains disjoint from the failed-screen support. Retain the side
and corner probes only BEYOND that support. If a junction has been
passed, the center of the affected side or corner stage lies within
O(v) of y, so a fixed-factor enlargement gives separation. Omitting
an entire side stage loses at most C (1 + v/d)^2; shifting or
omitting a corner stage loses at most C (1 + v/u)^beta.
Since the first possible failed radius is a fixed fraction of d,
sum_{v >= c_3 d} exp(-c v^2) (1 + v/d)^2 (1 + v/u)^beta
<= C d^(-2 - nu).
The retained central probe supplies A(b_*, R) throughout this case.
- For v >= c_2 R, DISCARD all arm probes and sum the absolute failure
probabilities, obtaining C exp(-c' R^2). This is bounded by the
RHS of (cl:boundary-kernel) BELOW: A(b_*, R) >= c R^(-g) and
d_* <= d <= u <= rho R make the reciprocal of that RHS at most a
fixed power of R.
This case includes EVERY failure whose support reaches the central
probe, because |y| >= R/100 and the central probe radius was chosen
much smaller than R/100. Thus no overlapping supports have been
treated as independent.
The conditional total-variation argument on screen success, together
with these failure bounds, proves uniformly:
| K_R(y) | <= C A(b_*, R) d^(-2 - nu) (d/u)^2 (u/R)^beta.
(cl:boundary-kernel)
Step 6 (dyadic summation). A dyadic class with distances comparable
to d, u has AREA at most C u d. Below the upper cap, it lies in corner
disks of radius C u and boundary strips of width C d.
- If u is capped, the total strip length is O(R) = O(u).
- If both distances are capped, the ambient region has area O(R^2)
= O(u d).
These assertions apply equally OUTSIDE the rectangle. The bottom classes
have the same bounds with d_* in place of a smaller distance. Summing
(cl:boundary-kernel) first over dyadic d <= u:
sum_{d <= u} C u d * d^(-2 - nu) (d/u)^2 (u/R)^beta
= (C/u) (u/R)^beta sum_{d <= u} d^(1 - nu)
<= C u^(-nu) (u/R)^beta,
where nu < 1 is used. The remaining dyadic sum is bounded by:
C R^(-beta) sum_{d_* <= u <= rho R} u^(beta - nu)
<= C R^(-xi_0) for every 0 < xi_0 < min{beta, nu}.
The strict inequality accommodates the logarithm when beta = nu.
Together with (calibration-comparison):
integral_{R/100 < |y| <= 10R} | K_R(y) | dy
<= C f(R) R^(-xi_0). (cl:boundary-integral)
Step 7 (far exterior, |y| > 10R). An agreement screen between the test
rectangle and y gives
| K_R(y) | <= C exp(-c |y|^2).
Its integral is at most C exp(-c R^2). The polynomial lower bound for f
following from (calibration-comparison) absorbs this into
C f(R) R^(-xi_0).
Combining (cl:bulk-integral) and (cl:boundary-integral), and decreasing
the exponent, proves (cl:tail-eq). QED.
----------------------------------------------------------------
42. Comparing the normalized derivatives
----------------------------------------------------------------
The tail estimate permits us to DISCARD insertions far from the origin.
For the remaining insertions we use the SAME interior messages in two
different rectangles. Rank-one approximation then controls a
DETERMINANT, so that no unknown leading factor needs to be identified.
LEMMA 42.1 (two-size determinant). Let nu' > 0 be the exponent in
Corollary 39.1. There are constants C, c > 0 such that, for sufficiently
large m, 100m <= R <= R' <= 2R, and almost every 0 < |y| <= m,
| K_R(y) f(R') - K_{R'}(y) f(R) |
<= C (m/R)^(2 lambda + nu') + C exp(-c m^2).
(cl:determinant-eq)
The constants are independent of the distance from y to the origin.
PROOF.
Step 1 (cuts and initial message data). Take concentric cuts about the
origin, starting at radius 10m and ending at a radius comparable to a
small fixed fraction of R, with consecutive radii in ratio L. Round the
final radius down to the last such scale. Its ratio to R is bounded
below by a positive constant depending only on L.
All cut collars lie strictly inside both rectangles and outside both
inserted points. If necessary, enlarge the fixed lower bound on R/m; the
remaining bounded ratios follow by increasing C, since all four scalar
quantities in the determinant are bounded.
Step 2 (screen reduction). First restrict every test to success of
EVERY cut screen, with the screens evaluated on the COMMON base process.
The probability discarded is at most
C sum_{j >= 0} exp(-c m^2 L^{2j}) <= C exp(-c' m^2).
Step 3 (initial messages). At the first cut:
a := signed connection-state message obtained by changing
the central color,
b_y := difference of this message WITH and WITHOUT the extra
fair point at y.
Both have ZERO MASS conditional on each seam. Their total variation
norms are bounded by 2 and 4 respectively. They depend on NEITHER R nor
R': on screen success their construction uses only the common interior
Poisson variables and the first seam. In particular this description
does not require disjoint neighborhoods of 0 and y.
Step 4 (transfer and boundary functionals). Let T be the composed
transfer through the intervening cuts. The exterior crossing tests in
R Q_0 and R' Q_0 define bounded linear functionals l_R and l_R' at the
final cut. The four restricted expectations are:
l_R(T a), l_R'(T a),
l_R(T b_y), l_R'(T b_y).
Their DETERMINANT is the evaluation of T^(x)2 on
b_y (x) a - a (x) b_y,
by l_R (x) l_R'. Corollary 39.1 therefore bounds it by
C (m / R)^(2 lambda + nu').
Step 5 (restore events). Restoring the discarded events changes the
determinant by at most C exp(-c m^2), because each scalar factor is
bounded. This proves (cl:determinant-eq). QED.
----------------------------------------------------------------
43. Summable logarithmic error
----------------------------------------------------------------
PROPOSITION 43.1 (summable logarithmic error). There are constants
alpha > 0 and C < infinity such that
| (R f'(R)) / f(R) + lambda |
<= C R^(-alpha) (cl:log-derivative-eq)
for all sufficiently large R.
PROOF.
Set g(R) = R f'(R) / f(R), defined and continuous for large R by Lemma
40.1 and (calibration-comparison). We prove a UNIFORM estimate on each
interval [R, 2R].
Step 1 (parameter choices). Fix
zeta = nu' / ( 4 (2 + 2 lambda + nu') ),
eta = nu' / 8,
m = R^zeta.
By the exponent estimate (exponent-eq), f(s) >= s^(-lambda - eta) for
all sufficiently large s.
Step 2 (near integral). For R <= R' <= 2R, divide (cl:determinant-eq) by
f(R) f(R') and integrate over |y| <= m:
| integral_{|y| <= m} K_R(y)/f(R) dy
- integral_{|y| <= m} K_{R'}(y)/f(R') dy |
<= C R^(2 lambda + 2 eta) m^2 [ (m/R)^(2 lambda + nu')
+ exp(-c m^2) ]
<= C R^(-nu'/2). (cl:normalized-near)
The power computation for the first term (using m = R^zeta):
exponent = -nu' + 2 eta + zeta (2 + 2 lambda + nu')
= -nu' + nu'/4 + nu'/4
= -nu'/2.
(The 2 lambda cancels against -2 lambda in the (m/R)^ power.) The
exponential term is smaller. For large R, the chosen m satisfies all
the scale restrictions of Lemmas 41.1 and 42.1.
Step 3 (tail). Lemma 41.1 bounds each complementary normalized integral
by C R^(-zeta xi).
Step 4 (dyadic oscillation). Since g(R) = 2 integral K_R(y)/f(R) dy by
(cl:derivative-eq), we get
sup_{R <= R' <= 2R} | g(R') - g(R) |
<= C R^(-alpha), alpha = min{ nu'/2, zeta xi } > 0.
(cl:dyadic-oscillation)
Step 5 (limit exists). The values g(2^n R_0) form a Cauchy sequence,
because the successive errors in (cl:dyadic-oscillation) are summable.
The same estimate between dyadic points shows that g(R) has a finite
limit gamma as R -> infinity, and summing along R, 2R, 4R, ... gives
| g(R) - gamma | <= C sum_{j >= 0} (2^j R)^(-alpha)
<= C' R^(-alpha).
Step 6 (identify gamma). Integration yields
log f(R) - log f(R_0) = integral_{R_0}^R g(s) ds / s.
Dividing by log R shows log f(R) / log R -> gamma. By (exponent-eq)
the same limit is -lambda. Thus gamma = -lambda, proving
(cl:log-derivative-eq). QED.
----------------------------------------------------------------
44. Completion of the amplitude theorem
----------------------------------------------------------------
PROOF OF THEOREM (Theorem 2, Part I).
Step 1 (integrated log derivative).
(d/dR) log ( R^lambda f(R) )
= lambda/R + f'(R)/f(R)
= O(R^(-1 - alpha)).
The right side is integrable at infinity. Since f(R) > 0 for large R,
the logarithm has a finite real limit. Consequently
R^(5/4) f(R) -> a in (0, infinity). (cl:amplitude)
Positivity and finiteness follow from convergence of the LOGARITHM,
rather than merely boundedness of the normalized probabilities. In fact
the argument also gives
R^(5/4) f(R) = a ( 1 + O(R^(-alpha)) ).
Step 2 (calibration). Recall
S_eps = f(1/eps),
m_eps = eps^(-2),
q_eps = ( E N_eps(Q_0) )^(-1).
The calibration limits (calibration-eq-a) and (calibration-eq-b) are:
m_eps q_eps S_eps -> c in (0, infinity),
q_eps E N_eps -> B in (0, infinity).
Equation (cl:amplitude) gives S_eps ~ a eps^(5/4). Hence
q_eps = ( E N_eps )^(-1)
= ( c / (m_eps S_eps) + o(1) ) * (1 + o(1))
= ( c / (eps^(-2) a eps^(5/4)) ) (1 + o(1))
~ (c / a) eps^(3/4).
Step 3 (conclusion). Therefore
eps^(3/4) E N_eps = eps^(3/4) / q_eps (1 + o(1))
-> B a / c =: c_V in (0, infinity).
Here N_eps counts COLOR-PIVOTAL SITES for the original unwired unit-square
event. The Poisson insertion was used only to differentiate f; it has not
changed the pivotal count in this last formula. Taking reciprocals
proves the equivalent assertion for the near-critical scale:
(E N_eps)^(-1) ~ c_V^(-1) eps^(3/4).
QED.
This completes the proof of the main theorem from Part I.
================================================================
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