editors: nobody
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BUFFERED COMPARISON AND STOPPING-BAND RESOLUTION IN CRITICAL ISING
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An expository, self-contained version, written to be readable by a
high-school student who is comfortable with probability and wants to see
the full argument, with no steps hidden.
The original paper is a LaTeX article. Here every theorem, lemma, and
corollary is restated in plain text and proved. Symbols are kept as in
the original, but each is defined the moment it first appears.
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PART 0. WHAT THIS PAPER PROVES, IN PLAIN WORDS
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The critical Ising model is a random colouring of the square grid by two
colours, call them + (up) and - (down). It is the model of a magnet, and
at one special temperature, beta_c, it has long-range structure: the
colours are correlated across huge distances, and interfaces (curves
separating + from -) behave like random curves with a well-defined
continuum limit.
Version 1 of this picture: explore (reveal) spins near an interface. You
discover a random jagged boundary. Later you care about spins FAR from
that boundary, on the other side of a fixed band of positive width. What
do those distant spins look like, given everything you revealed?
The visible boundary may have many thin "slits" (excursions that poke
toward you). Its signs may alternate. To say anything about the far side
you must, it seems, remember an infinite amount of jagged detail. This
paper proves that you do not. To a controlled total-variation error, the
distant spine law depends only on which of FINITELY MANY regular
("signed barrier") contours the visible boundary resembles, and the
barriers and error tolerances are chosen BEFORE the mesh gets fine. The
same approximation works uniformly for all bounded distant observables,
even ones that change with the mesh, and the descriptors (the finite
codes) converge in discrete total variation.
The engine has two halves.
Half A (comparison). Changing prescribed Ising spins on a set changes
any outcome on another set by at most a factor exp(4 arctanh(q0)), where
q0 is a zero-field random-cluster connection probability on the graph
with the intermediate prescribed spins deleted. On the critical square
lattice, q0 is tiny whenever a buffer of lattice separates the two sets,
so "far apart" means "almost independent". This works with arbitrarily
many common prescribed spins and mixed signs.
Half B (transmission). If two joint laws are ordered pointwise, the
effect of a small patch of changed boundary on a distant target is
bounded by the change of a normalized magnetization on an intermediate
"cut" contour. Summing over a cover of the cut turns the microscopic
comparison into a macroscopic total-variation statement on the target.
Then a geometric construction replaces the jagged visible boundary by a
regular signed contour with ordered upper and lower approximations, uses
incidence ordering to sandwich the actual transcript between them, and
uses transmission to convert the small mean gap into a total-variation
gap. A separable metric on boundary data plus a countable-cover argument
reduce infinitely many possible boundaries to finitely many barriers.
Everything rests on four imported truth-bricks, used as hypotheses and
cited each time:
(C1) RSW / crossing and one-arm estimates for critical FK-Ising
[Duminil-Copin, Hongler, Nolin 2011].
(C2) The Ding-Song-Sun correlation inequality for Ising with
arbitrary external fields [2023].
(C3) Bulk one- and two-spin limits and two-spin fusion for critical
Ising [Chelkak, Hongler, Izyurov 2021].
(C4) The critical Ising interface scaling limit is nested CLE_3, with
simultaneous common-edge approximation of ALL macroscopic loops
[Benoist, Hongler 2019].
Plus the Edwards-Sokal and Fortuin-Kasteleyn-Ginibre cluster
machinery and the Karlin-Rinott total-positivity theory.
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PART 1. THE ISING MODEL AND ITS RANDOM-CLUSTER FORM
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1.1 Finite Ising model. Let G = (V,E) be a finite graph. A spin
configuration is sigma in {-1,+1}^V. Given couplings K_e = K_uv >= 0 on
edges and fields h_v real on vertices, the Ising (Gibbs) law is
mu(sigma) = (1/Z) exp( sum_{uv in E} K_uv sigma_u sigma_v
+ sum_{v in V} h_v sigma_v ),
with Z = sum over sigma of the exponential (the partition function).
"Prescribing" a spin sigma_v = tau means fixing that coordinate and
dropping it from the sum, folding its incident interactions into the
fields of its neighbours. "Deleting" a vertex means removing it and all
its incident edges.
The model is FERROMAGNETIC: all K_e >= 0, so neighbours like to agree.
1.2 Pointwise comparison, stated. Fix three disjoint vertex sets I, J,
F, with I and J nonempty. Prescribe spins on F (the "common pins").
Compare the conditional law of the spins on I when the spins on J are
set to b versus when they are set to b'. The claim (Theorem in:comparison,
proved in PART 2) is that for every a,
exp(-4 arctanh q0) <= mu(sigma_I=a | sigma_J=b, sigma_F) /
mu(sigma_I=a | sigma_J=b', sigma_F)
<= exp( 4 arctanh q0).
Here q0 is the probability that the two contracted terminal sets are
connected in a zero-field random-cluster (FK) model on the graph with F
deleted. The bound is symmetric in I and J, and holds for mixtures too.
Because a density ratio lying in [e^{-L}, e^{L}] forces the total
variation distance of the two conditional laws to be at most
min(1, e^{L}-1), small q0 gives near-independence. arctanh q0 ~ q0 for
small q0, so the exponent is ~4 q0.
1.3 The random-cluster (FK) representation and the Edwards-Sokal
identity. For a finite graph with couplings K_e, put edge probability
p_e = 1 - e^{-2 K_e}. A bond configuration omega assigns each edge open
(1) or closed (0). The FK random-cluster measure with cluster weight 2
is
phi(omega) proportional to prod_e p_e^{omega_e}(1-p_e)^{1-omega_e}
* 2^{k(omega)},
where k(omega) is the number of connected components ("clusters") of
open edges, counting isolated vertices. Given omega, colour each cluster
with an independent fair +/- sign, EXCEPT clusters touching a prescribed
+ pin, which are forced +. The resulting spin law is exactly the Ising
law mu (Edwards-Sokal). Consequences we use repeatedly:
(a) mu(sigma_x sigma_y) of a zero-field law equals phi(x <-> y),
the probability that x and y lie in the same open cluster
(because two independent replicas "-" and "+" have a common open
path exactly when phi connects them; this is the Edwards-Sokal
identity used in PART 2).
(b) FK enjoys the FKG / positive association inequalities and the
domain Markov property; increasing edge sets and increasing
boundary wiring raise connection probabilities. These are the
Fortuin-Kasteleyn-Ginibre and Holley facts.
(c) Deleting prescribed spins only removes constraints, so restoring
them can only raise any connection probability in the comparator.
1.4 Total variation. For two probability laws nu, nu' on the same space,
||nu - nu'||_TV = sup_A |nu(A) - nu'(A)| = (1/2) sum |nu(x)-nu'(x)|.
If d nu' / d nu lies in [e^{-L}, e^{L}], then ||nu-nu'||_TV <= e^{L}-1
(the "minimum is at most one, maximum at least one" step). This is the
bridge from pointwise density bounds to law-level bounds.
1.5 The critical square lattice. From PART 3 onward we set every
retained coupling to the critical value
beta_c = (1/2) log(1 + sqrt 2),
take G a large square of the square lattice Z^2, and write
B_r = [-r, r]^2 intersect Z^2, Q_r = {||z||_inf < r} (open square), and
gamma_r = boundary of Q_r. The random-cluster edge probability is
p_c = 1 - e^{-2 beta_c}. All constants are independent of the number of
vertices.
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PART 2. POINTWISE COMPARISON ON A FINITE GRAPH
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Goal: prove Lemma fd:ratio, the pointwise density comparison, from the
following rectangle lemma. Both are due to the authors; here is the
complete argument.
2.1 Log-supermodularity. On a product of two-point spaces {-1,+1}^n, a
positive weight p satisfies the LATTICE CONDITION (is log-supermodular /
"Totally Positive of order 2") if
p(x and y) p(x or y) >= p(x) p(y) for all x, y,
where "and"/"or" are coordinatewise min/max. For a ferromagnetic Ising
weight, every pair coupling K_uv sigma_u sigma_v contributes a factor
that is log-supermodular and every field h_v sigma_v a factor with zero
mixed difference, so the total weight is log-supermodular.
2.2 Lemma fd:mtp2 (extreme rectangles in a ferromagnetic marginal).
Let p(a,b) > 0 be a marginal of a finite ferromagnetic Ising law, on
two disjoint blocks of spins. Write f = log p and
L = f(+,+) + f(-,-) - f(+,-) - f(-,+).
Then for all configurations a, a', b, b':
| f(a,b) + f(a',b') - f(a,b') - f(a',b) | <= L. (*)
Proof. Step 1: summing out spins preserves log-supermodularity. It
suffices to check one binary coordinate being summed out, on a square of
two other coordinates. Call the eight un-summed weights a,b,c,d on one
"slice" (ordered --,+-, -+, ++) and e,f,g,h on the other slice. The
lattice condition gives
ad >= bc, eh >= fg, ah >= bg, ah >= cf.
Multiply the first and second: (ah)(ed) = (ad)(eh) >= (bc)(fg) so
ah*ed >= bg*cf. Now a tiny numerical fact: if X >= max(U,V) and XY >= UV
then X+Y >= U+V. (Assume U >= V. Then X + UV/X >= X + V >= U + V since
X >= U, using UV/X <= V.) Apply it with (X,Y,U,V) = (ah, ed, bg, cf):
ah + ed >= bg + cf.
Add ad >= bc and eh >= fg to both sides:
(a+e)(d+h) = ad + ah + ed + eh
>= bc + bg + cf + fg = (b+f)(c+g).
The left side is (slice_one summed)(slice_two summed); the inequality is
exactly log-supermodularity of the summed pair. So the two-block marginal
is log-supermodular. For more coordinates, eliminate one binary coordinate
at a time (a nonnegative adjacent-coordinate square difference is
equivalent to log-supermodularity on a Boolean cube, by summing along
paths), concluding p is log-supermodular.
Step 2: reduce any rectangle to the extreme one. For ordered u <= v and
s <= t put
Delta(u,v; s,t) = f(u,s) + f(v,t) - f(u,t) - f(v,s) >= 0,
the (nonnegative, by log-supermodularity) mixed difference. Let
D = f(a,b) + f(a',b') - f(a,b') - f(a',b).
The exact telescoping identity
L - D = Delta(-,a'; -,b) + Delta(a,+; b',+)
+ Delta(a',+; -,b') + Delta(-,a; b,+)
holds (expand: every f-entry appears with the coefficient it has in L-D;
this is a four-corner add-and-subtract on the rectangle). Each Delta is
>= 0, so L - D >= 0, i.e. D <= L. Swapping b and b' replaces D by -D,
so -D <= L too, and |D| <= L. No order between a,a' or between b,b' is
needed. This proves (*). []
Remark. Restricting both terminal blocks to a single common sign is
exactly "contracting" each block to one vertex (all internal edges become
constant shifts; parallel edges to the outside add up, still >= 0).
So the extreme rectangle is the log-odds rectangle between the four
constant terminal configurations.
2.3 Lemma fd:ratio (pointwise ratios with arbitrary common pins).
Let G be a finite ferromagnetic Ising graph, I, J disjoint nonempty
vertex sets, F a third set with prescribed spins tau. Delete F (retaining
its effective fields on the neighbours), and contract I to one vertex and
J to one vertex. Let q0 be the zero-field FK connection probability of
the two contracted vertices in this contracted, F-deleted graph (with any
additional external fields set to zero). Then for all a and all b,b' on J,
exp(-4 arctanh q0) <=
mu(sigma_I=a | sigma_J=b, sigma_F) /
mu(sigma_I=a | sigma_J=b', sigma_F) <= exp(4 arctanh q0).
The same holds with either conditional law replaced by a mixture, and
with arbitrary extra external fields present (they are zeroed in q0).
Proof. Step 1: effective fields and the two-spin reduction. Each pin
tau_v (v in F) contributes K_uv tau_v to the field at its surviving
neighbour u; integrate these in and delete F. Now apply Lemma 2.2 to the
joint marginal on (I,J). Restricting a terminal block to a constant sign
is its contraction, so we get a two-spin marginal. Its unnormalised
weights have the form
w(s,t) = exp( C + K s t + A s + B t ), s,t in {-1,+1},
with K = L/4 >= 0 (L the extreme rectangle, nonnegative by Step 2 of
2.2), and A, B absorbing all remaining effective fields.
Step 2: balance the fields. Add external fields -A at the I-vertex and
-B at the J-vertex. This multiplies w by e^{-As-Bt}, giving weights
proportional to e^{K s t}: a zero-mean two-spin law. For a symmetric
two-spin law with interaction K, the means vanish and the covariance is
<sigma_I sigma_J>_0 = tanh K.
Step 3: the Ding-Song-Sun covariance inequality. Corollary 1.3 of
[Ding-Song-Sun 2023] (our brick C2) says that for ANY finite ferromagnetic
Ising graph with ARBITRARY real (possibly mixed-sign) fields and two
disjoint vertex sets I,J, the two-point covariance obeys
<sigma_I sigma_J> <= <sigma_I sigma_J>_0,
where the right side is the covariance at zero fields. Apply it: the
left side is tanh K, the right side, by Edwards-Sokal, is exactly q0.
(Contracting a terminal, adding parallel edges, or adding internal
constant edges does not destroy ferromagnetism or the hypothesis of C2.)
tanh K <= q0, hence K <= arctanh q0, hence L = 4K <= 4 arctanh q0.
Step 4: from the extreme rectangle to oscillation. Fix b,b'. Lemma 2.2
bounds, for every a and a', the change of log p(a|b) - log p(a|b') by
L, because that is the rectangle with the J-coordinate switched. So the
log of the density ratio a -> p(a|b)/p(a|b') oscillates by at most L.
That ratio has mean one under the denominator law p(.|b'); a random
variable with mean one and log-oscillation <= L is trapped in
[e^{-L}, e^{L}] (its minimum is <= 1 <= its maximum). Exponentiating with
L <= 4 arctanh q0 gives the displayed bounds. Averaging a pointwise
inequality preserves it, giving the mixture statement. []
2.4 Remark fd:symmetric-terminals. The extreme rectangle L and q0 are
symmetric under swapping I and J: the two contracted vertices are
interchangeable in the FK comparator. So comparison holds with observed
and prescribed terminals reversed, which we use later when a small
changed patch is the "source" and a large observed set is the "target".
Common pins of opposite sign are never wired together; they are simply
deleted from the comparator.
2.5 Restating Theorem in:comparison. It is exactly Lemma 2.3 with the
role assignment "I = the observed terminal, J = the changed terminal,
F = arbitrary common pins". Its statement in PART 0 / PART 1 is the
advertisement; the proof is above. The whole rest of the paper spends its
time making q0 small in specific geometries.
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PART 3. CRITICAL COLLARS: BUFFERED AND SMALL-PATCH BOUNDS
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We now turn Lemma 2.3 into geometric bounds on the critical square lattice.
Recall p_c = 1 - e^{-2 beta_c}, cluster weight 2, and FK "open" always
means open on actual lattice edges (a mere boundary identification is not
an open path).
3.1 Critical crossing input. Brick C1 ([Duminil-Copin, Hongler, Nolin
2011], arXiv v1 Theorem 1) states: a critical FK-Ising lattice rectangle
whose two side lengths are at least 1 and whose aspect ratio lies in
[A^{-1}, A] has probability at least a constant c_A > 0 of either
specified open crossing. By planar duality the same holds for either
specified DUAL crossing.
3.2 Lemma fd:conditional-rsw (conditional crossing tests). Summary of
what we need from C1, in the form the paper uses.
(i) For any boundary wiring (any partition of the boundary that the
exterior edges induce), the open- and dual-crossing probabilities
of a fixed aspect-ratio rectangle are bounded below by c_A. Changing
each side by O(1) lattice spacings only changes c_A.
(ii) A DECREASING dual-crossing test in a vertex-deleted graph still has
probability >= c_A, conditionally on all bonds OUTSIDE its full
rectangular test region, provided no terminal identification has a
vertex in the interior of that region. Extra vertex deletions keep
the bound.
(iii) For finitely many such tests on DISJOINT sets of unexposed edges,
the conditional lower bounds multiply.
Proof of (ii)+(iii). (ii) The FK domain Markov property turns any
exterior bond configuration into a boundary wiring. Restore the deleted
interior vertices (isolated first, then their edges) and wire the whole
outer boundary. The restored measure stochastically dominates the
conditional original one, because a single edge's conditional connection
probability is p_c (endpoints not yet joined) or p_c/(p_c+2(1-p_c))
(endpoints already joined through the wiring), both at least p_c, and
adding edges or wiring can only raise them. A dual crossing is a
DECREASING event, so its probability in the original law is at least its
probability in the dominating wired law, i.e. >= c_A. The "no terminal in
the interior" clause guarantees we are not discarding an identification
that was helping. (iii) Order the test regions; conditioning on the next
region's exterior includes all earlier results, so multiply by induction.
The companion product UPPER bound for simultaneous failure follows the
same way from the union bound. For a side forced too short by rounding,
finitely many prescribed open/closed edges using the same one-edge
probabilities supply a uniform replacement. []
3.3 Dual-circuit corollary (used in both bounds below). An annulus of
fixed radius ratio surrounding a small disk contains a DUAL circuit with
probability bounded below, uniformly in the boundary wiring. Construction:
cover a middle sub-annulus by finitely many overlapping aspect-ratio
rectangles, demand dual crossings in both directions of each overlap
square (Lemma 3.2), so the crossings join into a closed dual loop that
separates the terminals even after their separate internal wiring. []
3.4 Corollary fd:buffered-ratio (buffered comparison). Fix c > 0. There
is K_c < infinity such that, on the critical square lattice, if
I is contained in z + B_R and J is contained in the complement of
z + B_{(1+c)R}, then every conditional density ratio of Lemma 2.3 lies in
[K_c^{-1}, K_c]. Other common pins may sit anywhere off I union J.
Proof. The annulus between z+B_R and z+B_{(1+c)R} has fixed radius ratio,
so by 3.3 it contains a dual circuit with probability >= c_1(c), uniformly
in the wiring. In the FK comparator of Lemma 2.3, restore the deleted
common pins (this only helps the connection), so the connection
probability satisfies q0 <= 1 - c_1(c). Hence arctanh q0 <= arctanh(1 -
c_1(c)) = a geometric constant, and Lemma 2.3 gives pointwise ratio in
[e^{-L}, e^{L}] with L = 4 arctanh q0 <= some K_c'. Set K_c = e^{L}. []
3.5 Corollary fd:patch-ratio (small patch vs. remote set). There are
absolute constants K < infinity and delta > 0 so that, if I is contained
in z + B_w, J is contained in the complement of z + B_R, and
R >= 8w >= 8, then the logarithm of every such conditional density ratio
is at most K (w/R)^{delta} in absolute value. In particular the total
variation distance is at most K' (w/R)^{delta}, and the expectation of a
bounded test function of sup-norm ||h||_inf differs by at most
2 K' ||h||_inf (w/R)^{delta}. NO free neighborhood is required around the
small patch I.
Proof. Between B_{2w} and B_{R/2}, cut out disjoint annuli each of fixed
radius ratio (say 4) and with disjoint sets of unexposed test edges. There
are at least c_2 log(R/w) - c_3 of them. If the two contracted terminals
were FK-connected, the connection would have to avoid the guaranteed dual
circuit in every annulus, so by the product bound Lemma 3.2(iii),
q0 <= C_1 (w/R)^{delta_1} for some absolute delta_1 > 0.
Additionally the fixed annulus between I and J gives q0 <= 1 - c_4
(bounded away from 1). On the interval [0, 1 - c_4] we have
arctanh q <= C_2 q, so
L/4 = arctanh q0 <= C (w/R)^{delta}
for a new absolute delta in (0, delta_1], and Lemma 2.3 gives the log
bound. Total variation and expectation bounds then follow by integrating
|density ratio - 1| against the denominator law (||h||_inf factor). []
3.6 Corollary fd:finite-cover (local charts and finite patch covers).
Facts extending 3.4-3.5 to charts and to partitions.
(a) Both bounds survive a chart clipped by a free or a fixed physical
boundary, as long as the local graph and the common prescribed
boundary spins agree in the two compared laws. Deleted vertices and
missing edges are allowed in the chart (restore them in the FK
comparator; decreasing dual tests keep their bound).
(b) For laws on DIFFERENT domains (plane, torus, cylinder), the buffered
comparison holds whenever their common collar is an isometric
square-lattice chart and every differing datum lies beyond it; this
is the mixture statement of Lemma 2.3, applied by conditioning on the
boundary of the common chart.
(c) If the observed set splits into at most m patches, each separated
from every changed datum by a fixed positive relative distance, then
the whole observed law has a pointwise density comparison with
constant equal to the PRODUCT of the patch constants. The same holds
if instead the changed source set is so partitioned away from the
whole observed set.
Proof. (a)-(b): fix a common finite local conditional graph, restore in
its FK comparator any missing square-test vertices/edges (this only
raises connectivity, and decreasing dual tests still bound q0), then use
Lemma 2.3; for different volumes, both laws are mixtures of the same
local kernels, so the mixture clause applies. (c): reveal the observed
patches in a fixed order and apply the chain rule to factor the
configuration probability; for the next patch every earlier revealed
patch is a common pin (they may lie anywhere off the two terminals), each
conditional ratio is bounded by its patch constant, and multiplication
gives the product. For a source cover, telescope the changes and use
Remark 2.4 to place the small patch in the inner terminal. Mixtures over
source assignments and common randomizations preserve the bounds. []
Interpretation. On the critical lattice, "separated by a buffer" already
forces near-independence of the conditional laws, with every common pin
tolerated. The only price for a narrow small patch is a power law in the
ratio of scales.
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PART 4. SPIN CORRELATIONS AND MAGNETIC MOMENTS
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Throughout, the Ising graph is the critical nearest-neighbour square
lattice with no external fields except those induced by prescribed spins;
pins beyond the stated free neighborhoods may have any signs.
4.1 Lemma fd:static-correlations (static correlation inputs). With
constants depending only on the indicated fixed bulk separations and
aspect ratios:
(1) ARM BOUNDS. In a square of radius r, under ANY FK boundary wiring,
for x in a fixed central subsquare,
c r^{-1/8} <= phi(x <-> boundary of B_r) <= C r^{-1/8}.
In the plus Ising law this is the magnetization estimate.
(2) PLUS COVARIANCE. In a plus rectangle of scale r and fixed aspect
ratio, for two distinct points in separated fixed compact bulk sets,
c r^{-1/4} <= Cov_+(sigma_x, sigma_y) <= C r^{-1/4}.
(3) PLANE TWO-POINT. The infinite-volume zero-field law on Z^2 is unique
and spin-flip symmetric, and for one lattice constant c_sigma > 0,
E(sigma_x sigma_y) = c_sigma |x-y|^{-1/4} (1+o(1)) as |x-y| -> oo,
0 <= E(sigma_x sigma_y) <= C (1 + |x-y|)^{-1/4},
the asymptotic uniform in direction.
Proof.
(1) The central-subsquare lower bound is [DHN 2011, arXiv v1 Lemma 26].
For another bulk point x', take a square of radius c' r about x' inside the
original square; its free-boundary arm to the smaller square is still
available inside the larger domain. Require additionally an open circuit
in the outer half-annulus of that smaller square and a rectangle crossing
from its inner half out to the original boundary. The arm, circuit, and
crossing then join, giving x' <-> boundary of the big square. RSW gives
the circuit and crossing jointly a fixed positive probability, and FK
association multiplies this by the arm lower bound. The upper bound uses
the wired arm in a contained square. As the boundary recedes the
magnetization tends to zero.
Uniqueness and symmetry (part of (3)): couple the plus-boundary and
minus-boundary laws monotonically (the disagreement probability equals
half their mean difference at a vertex). A union bound over any fixed
finite set forces the two extremal infinite-volume limits to agree, so the
infinite-volume law is unique, and symmetry follows from the +/- exchange.
(2) Apply bulk one- and two-spin convergence [CHI 2021, Theorem 1.3]
(brick C3) in the fixed rescaled rectangle. Strict positivity of the
limiting connected function: in the upper half-plane, let
q = |z-z'| / |z - conj(z')| in (0,1). The plus one- and two-spin formulae
[CHI 2021, Theorem 7.1 and Eq. (7.3)] give
<sigma_z sigma_z'>_+ / (<sigma_z>_+ <sigma_z'>_+)
= ( (q^{1/2} + q^{-1/2})/2 )^{1/2} > 1.
Conformal covariance transfers this strict inequality to a rectangle.
Continuity and compactness give a positive MINIMUM of the connected
function on separated compact bulk sets. Uniform bulk convergence turns
this into a discrete lower bound for large r; finite energy absorbs the
finitely many smaller scales. The upper bound also follows from
convergence, or from two disjoint wired arm tests.
(3) Put R = |x-y|, u = (y-x)/R, and bracket the plane two-point function
between the free and plus laws in a square of radius T R centred at x.
Fix T > 2 first, then let R -> oo. Theorem 1.3 of [CHI 2021] gives the
continuum limits F_T^b(0,u), b in {free,+}, uniformly for |u| = 1.
Conformal scaling gives F_T^b(0,u) = T^{-1/4} F_1^b(0, u/T). The
normalization and two-spin fusion [CHI 2021, Prop. 5.3, Thm. 6.3,
Eq. (6.12)] give F_1^b(0, u/T) = T^{1/4}(1 + o_T(1)) for both boundary
conditions; Remark 6.4 there makes the error uniform in direction
(colliding pair in the bulk, no other insertions). Letting T -> oo after
the fixed-T mesh limit proves the plane asymptotic with common lattice
constant c_sigma, uniform in direction. The uniform upper bound: a
connection of x to y forces arms in the two disjoint boxes of radius
floor(|x-y|/4); condition on all other edges and use the wired arm bound.
Nonnegativity follows from the FK representation. []
4.2 Lemma fd:fk-product (product arm bound). Let Q be a lattice square
with an ordinary FK bond law under arbitrary boundary wiring. After
sampling bonds, colour every cluster touching the boundary by any joint
sign law conditional on the bonds, and give every interior cluster an
independent fair sign; (non-boundary cluster signs are then mutually
independent and independent of the boundary-cluster signs). Suppose
x_1,...,x_n (n >= 2) lie a distance at least c s from the boundary of Q
and inside a set of diameter at most A s. With
r_i = max(1, floor( min( c s / 4 , (1/4) min_{j != i} ||x_i - x_j||_inf ) ) ),
we have
| E prod_{i=1}^n sigma_{x_i} | <= C_{n,c} prod_{i=1}^n r_i^{-1/8}.
The bound also holds conditionally on arbitrary bonds exterior to Q
(which induce only a boundary wiring on Q).
Proof. First cancel insertions occurring an even number of times at one
vertex (a repeated index has r_i = 1, so dropping its factor costs
nothing; all remaining r_i keep their original values). Now, at each
remaining vertex, the conditional cluster-sign average is nonzero only if
its free cluster contains another odd insertion, i.e. it needs an OPEN
ARM to the boundary of its own ball of radius r_i (a cluster reaching a
prescribed-sign boundary needs the same arm). The balls of radius
r_i > 1 are disjoint and lie inside Q. Their arm probabilities,
conditionally on all exterior edges, are at most their fully-wired
probabilities, which Lemma 4.1(1) bounds by C r_i^{-1/8}. Sequential
conditioning multiplies these bounds, and absolute cluster-sign averages
are at most one; radius-one balls and parity cancellations only remove
requirements. Exterior bond conditioning supplies only boundary wiring,
so the same argument applies conditionally. []
4.3 Lemma fd:moments (buffered line and area moments). Fix A, c > 0, an
integer p >= 1, and nu in {1,2}. Let S be a set of lattice points of
diameter at most A s with
|S intersect B_t(x)| <= A (t+1)^nu (t >= 0), |S| <= A s^nu.
Assume the Ising law has a FREE square-lattice neighborhood of width c s
about S (arbitrary pins beyond it). Then for weights |a_x| <= 1,
E | sum_{x in S} a_x sigma_x |^{2p} <= C_{p,A,c} s^{2p(nu - 1/8)}.
This covers a fixed number of line segments or square contours (nu = 1)
and area sums (nu = 2); it holds in a buffered chart of a torus or
cylinder, and there is no pin on S.
Proof. Step 1: reduce to a plus square with buffer. Use Corollary 3.4
(finitely many times if necessary) to compare to the plus law in a
concentric square with proportional buffer. It suffices to bound the
plus law there. Expand the even power and take absolute values of the
spin-product expectations. Lemma 4.2 with the diameter bound reduces the
sum to a constant times
sum_{x_1,...,x_{2p} in S} prod_{i=1}^{2p}
( sum_{j != i} (1 + ||x_i - x_j||_inf)^{-1/8} ). (G)
(Here we replace the capped nearest distance of 4.2 by the uncapped one;
since all points lie within diameter A s, the capped value is a fixed
multiple from below of the uncapped value plus one.)
Step 2: combinatorial expansion. Put q = 1/8. Expand the product in (G)
by choosing, for each index i, one neighbour j(i) != i. The resulting
directed graph on the 2p indices has outdegree one, so each connected
component is an oriented CYCLE of length >= 2 with in-trees attached.
The counting hypothesis gives, uniformly in y,
sum_{x in S} (1 + ||x - y||_inf)^{-q} <= C s^{nu - q},
sum_{x in S} (1 + ||x - y||_inf)^{-2q} <= C s^{nu - 2q},
both by dyadic distance shells (q and 2q are less than nu).
Count one component of m vertices. Summing an in-tree leaf removes one
edge and costs C s^{nu - q}. Once only the cycle remains, sum one vertex
incident to two cycle edges by Cauchy-Schwarz, costing C s^{nu - 2q}
(for a 2-cycle the two edges share endpoints, and the second displayed
bound applies). The remaining graph is a path: remove leaves at
C s^{nu - q} each, and the last vertex at C s^{nu}. So a component of m
vertices costs at most C_m s^{m(nu - q)}. There are at most
(2p - 1)^{2p} neighbour choices. Multiplying over the components (whose
vertex counts sum to 2p) gives
(G) <= C s^{2p(nu - q)} = C s^{2p(nu - 1/8)}.
Step 3: large or bent support. Partition S by overlay squares of side
c s / 16; at most C(1 + A/c)^2 meet S. In each nonempty piece pick
z in S in that square. The piece lies in z + B_{c s/8}, and
z + B_{c s/2} lies in the assumed free neighbourhood (z is in S). Compare
just this local marginal to the plus law in z + B_{c s/2}; the counting
hypotheses persist with constants depending only on A, c. Apply the bound
of Steps 1-2 to each piece and use the triangle inequality in L^{2p}.
Lattice rounding at bounded c s is absorbed in the constants. The same
construction works in periodic charts. []
4.4 What these three lemmas buy. Lemma 4.3 gives a uniform second-moment
(and higher) bound for a normalized magnetization on a line or contour
of scale s, with exponent nu - 1/8. In PART 5 the field
M_J = s^{-7/8} sum_{x in J} sigma_x (nu = 1, so exponent
2*(1 - 1/8) = 7/4 in the second moment, i.e. a norm of order
s^{7/8} / s^{7/8} = 1 after normalization) is the "magnetic" quantity
whose ordered change will control transmission.
--------------------------------------------------------------------------------
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PART 5. ORDERED TRANSMISSION ACROSS A CUT (5A: the doubled model)
--------------------------------------------------------------------------------
Pointwise comparison (PART 2-3) controls one small patch. Transmission
turns a change spread along a whole separating cut into a total-variation
bound on a distant target, priced by the change of a normalized
magnetization on the cut. The tool is a DOUBLED Ising model and the FK
bonds of its two clones.
5.1 Geometry: admissible cut. A source S and a target T are disjoint
vertex sets; common pins off S union T are allowed. Delete the common pins
(retaining S as boundary), and let J be a finite set, disjoint from S, T,
and the common pins, that SEPARATES S from T: every path from S to T meets
J. Then the continuation on the target side of J does not depend on the
prescribed source spins.
Definition tr:admissible (an admissible cut at scales w, s). Fix
constants A, K >= 1 and a in (0, 1/(16A)). For 0 < w <= s, a separating
cut J is admissible if it splits into patches b_i with associated straight
counting segments L_i inside J, centres z_i and scales
ell_i in [a w, A w], such that:
(1) b_i union L_i is inside z_i + B_{ell_i} and |L_i| in [a ell_i, 2 ell_i];
(2) each z_i + B_{16 ell_i} is a pin-free planar lattice chart (free also
of source pins);
(3) each site of J lies in at most K counting segments;
(4) T is at distance >= s/A from every z_i, 16 ell_i <= s/A, and the
patch comparison of Corollary 3.5 applies up to a fixed multiple of s.
Constants may depend on a, A, K and the chart, but not on w, s, the number
or values of source pins, or the mesh. Concentric square contours with a
free collar of width ~ w, and parallel periodic rows, are the main examples.
5.2 Magnetization on the cut. Write
M_J = s^{-7/8} sum_{x in J} sigma_x. (5.2)
The normalization always uses the reference scale s, not the patch scale.
Proposition tr:transmission. There is an exponent delta in (0, 1/8),
obtained by shrinking the delta of Corollary 3.5 if needed, with the
following property. Prescribe two ORDERED configurations on S (one pointwise
>= the other), keep identical common pins, and let Delta denote the
higher-law minus lower-law expectation. For an admissible cut at scales
w, s, a bounded observable h supported on T, and w large,
| Delta h | <= C ||h||_inf (w/s)^{-7/8 + delta} Delta M_J. (5.3)
Equivalently, if ||h||_inf <= H, then Delta M_J >= c_H (w/s)^{7/8 - delta}
|Delta h|. Constants are uniform as w/s decreases within the geometry.
Why the exponent. A changed patch of diameter ~ w moves the conditional
expectation of h by at most C||h||_inf (w/s)^delta (patch comparison). The
doubled coupling gives: conditional on a disagreement reaching that patch,
the expected number of disagreements on its counting segment is at least
c w^{7/8}. Bounded overlap sums these over J; the normalization s^{-7/8}
yields (w/s)^{-7/8+delta}. We now prove the second (local mass) estimate,
which is the hard part, then assemble.
5.3 Two ordered replicas. Start on a finite graph G = (V,E) with couplings
K_e >= 0 and pin set P. For a retained FREE set H (in V minus P) and
couplings J_e, let Z_{P,tau}(H) be the Ising partition sum on P union H
with signs tau on P (edges between pins contribute constants). In our
planar application the original couplings are K_e = beta_c; the FK bonds
below use DOUBLED couplings on a random induced set, so the critical FK
crossing theorem is not applied directly to them.
Take two independent Ising configurations eta, xi whose signs on the SAME
pin set P are ordered, with eta the higher. Let U (subset of P) be the
DISAGREEMENT pins and A = P minus U the AGREEMENT pins, whose common signs
are tau. Let W = V minus P, and let G (subset of W) be the free sites where
the two ORIGINAL replicas disagree. The full auxiliary disagreement set is
G union U.
Lemma tr:doubled-coupling. With partition sums at couplings J_e = 2 K_e,
the law of G is
mu_tau(G) proportional to Z_{U,+}(G) Z_{A,tau}(W minus G). (5.4)
Conditional on G, the pair of spins eta = -xi on G union U is the first
law on the right of (5.4), independent of the common spins on the
complement. Put FK bonds omega on G union U. There is a coupling (eta, xi)
of the two Ising marginals with eta >= xi and
eta_x - xi_x = 2 * 1{x <-> U},
E eta_x - E xi_x = 2 P(x <-> U). (5.5)
A connection to U uses actual open edges, and is false for x not in
G union U.
Proof. For an edge with both ends inside G union U, or both ends outside,
the two replicas' energies add to twice the energy of their common
orientation; for a cross edge they cancel. This gives (5.4) and the
conditional independence. Edges between two pins are constants in G, so no
exception is caused.
Colouring: every FK cluster of G union U that avoids U carries an
independent fair orientation. On each such cluster replace the pair of
opposite spins by two copies of the first replica's sign: the first
replica is unchanged, and the second replica's fair cluster signs are
merely negated, so its conditional marginal is unchanged. Clusters meeting
U keep the pair (+,-); agreement sites keep equal signs. This gives eta >=
xi and (5.5). The auxiliary set G need not equal the final disagreement
set; the latter is exactly the union of clusters meeting U. []
5.4 Increasing order and the plus-agreement reference. Order pairs
(G, omega) by coordinatewise inclusion, padding omega by closed edges off
G union U; call an event or function increasing in this order. The
PLUS-AGREEMENT REFERENCE changes all signs on A to plus, keeping the
disagreement pins U. The coupled replicas now differ exactly on clusters
meeting U. The density of the mixed-agreement law relative to this
reference is, up to normalization,
r(G) = Z_{A,tau}(W minus G) / Z_{A,+}(W minus G). (5.6)
Lemma tr:site-gain (spin products and adding a vertex). Fix pins P,
couplings J_e >= 0; H is the retained free set.
(1) With all pins plus, every spin-product expectation is nonnegative and
increases when couplings increase, when free vertices and their
ferromagnetic edges are ADDED, or when extra spins are pinned plus
(the product involves only pre-existing vertices).
(2) For every finite Q subset of P union H,
| E_{P,tau,H} prod_{x in Q} sigma_x |
<= E_{P,+,H} prod_{x in Q} sigma_x. (5.7)
(3) H -> Z_{P,+}(H) is log-supermodular. For v not in P union H the site
gains obey
Z_{P,tau}(H union {v}) / Z_{P,tau}(H)
<= Z_{P,+}(H union {v}) / Z_{P,+}(H). (5.8)
Proof. Use FK with p_e = 1 - e^{-2 J_e}. An ACTUAL cluster is a component
of open edges of the original graph (virtual wires between pins do not
count). Let k_0(omega) be the number of actual clusters avoiding P. The
all-plus bond weight is proportional to
prod_e p_e^{omega_e}(1-p_e)^{1-omega_e} 2^{k_0(omega)}, (5.9)
i.e. the FK law with all pins wired; clusters touching a pin are +, the
rest are fair.
Even-parity event: let F_Q be the event that each actual cluster avoiding
P contains an even number of vertices of Q. Conditional sign averaging
gives
E_{P,+,H} prod_{x in Q} sigma_x = phi_{P,+,H}(F_Q). (5.10)
F_Q is INCREASING: merging two even unpinned clusters stays even; merging
any cluster with a pinned cluster removes its parity restriction; adding
an isolated vertex off Q adds an even (empty) parity. So FK monotonicity
under edge/parameter/wiring increase proves (1). (Pinning plus is the same
mechanism; positive fields can be modelled as edges to one extra plus
vertex.)
Mixed signs: put D_tau = {no actual cluster contains two opposite-sign
pins}. The mixed-sign bond weight is (5.9) times 1_{D_tau}; so its law is
phi_{P,+,H}(. | D_tau). (An edge between opposite pins must be closed.)
The conditional sign average has absolute value at most 1_{F_Q}, so by
association of the wired FK law,
| E_{P,tau,H} prod_{x in Q} sigma_x |
<= phi_{P,+,H}(F_Q | D_tau) <= phi_{P,+,H}(F_Q),
proving (5.7).
Site gain. Summing the new spin v gives the exact identity
g_{P,tau,H}(v) = Z_{P,tau}(H union {v}) / Z_{P,tau}(H)
= 2 E_{P,tau,H} cosh( sum_{u in P union H} J_vu sigma_u ). (5.11)
With neighbour set N, the cosh expands as
cosh(sum_{u in N} J_vu sigma_u) = prod_{u in N} cosh J_vu *
sum_{Q subset N, |Q| even} prod_{u in Q} tanh J_vu prod_{u in Q} sigma_u, (5.12)
all coefficients nonnegative. Under plus pins, enlarging H increases the
old product expectations, adds nonnegative terms, and raises the prefactor,
so g_{P,+,H}(v) increases with H. Increasing gains are equivalent to
log-supermodularity of H -> Z_{P,+}(H) (add the vertices of H_1 minus H_2
one at a time). Finally (5.7) term by term in (5.12) gives (5.8). []
Lemma tr:increasing-conditioning. The plus-agreement reference of
(G, omega) is positively associated. The mixed-agreement law has an
increasing density, depending only on G, against this reference. Hence for
any increasing event E of positive probability and any increasing f,
E_tau[f | E] >= E_+[f]. (5.13)
No restriction on number or placement of pins.
Proof. By Lemma 5.4, both factors of (5.4) with plus pins are
log-supermodular in G; complementation preserves that, so their product
is log-supermodular and the reference set law is positively associated.
Conditional on G the FK law is associated and stochastically increases
with G (add isolated vertices, then raise edge parameters from zero), so
expectations of increasing functions increase with G; decomposing a
covariance into conditional covariance plus covariance of conditional
means proves joint association. The mixed density r(G) of (5.6) increases
with G by (5.8); the conditional law in (5.13) has density proportional to
r * 1_E against the associated reference, which is increasing, and
association finishes. No association of the mixed law itself is needed. []
Proposition tr:local-domination. Let Q subset of W contain no pins. In Q
form the doubled model with PLUS signs on every exterior neighbour of Q
and the original couplings K_e; call its auxiliary set-and-bond law
P_Q^{++}. Then for every increasing f of the auxiliary set and bonds
INTERNAL to Q, and every increasing event E in the original doubled model,
E_tau[f | E] >= E_Q^{++}[f]. (5.14)
(the sure event allowed).
Proof. By (5.13) it suffices to compare the plus-agreement reference with
P_Q^{++}. Fix v in Q and the membership of all other sites of Q. With D
the full disagreement vertices except v (pins included), H the full
agreement vertices except v, the odds of v in G are
mu_+(v in G | rest) / mu_+(v not in G | rest)
= g_{U,+,D minus U}(v) / g_{A,+,H minus A}(v). (5.15)
The numerator (adding exterior disagreement vertices, edges, plus pins)
is at least the induced disagreement gain in Q. For the denominator, first
pin every exterior agreement vertex PLUS (this raises the gain by the
parity/expansion argument), then restore every missing exterior neighbour
of Q as a plus pin (again raises it); the result is exactly the
agreement-site gain of the box-alone model (exterior-only factors cancel
in the site ratio). Thus (5.15) dominates the box-alone odds at every
fixed interior configuration, so the reference interior-set law has an
increasing density against the box-alone set law, which is associated by
Lemma 5.4; tilting an associated measure by an increasing density gives
stochastic domination of the sets. Given the set, conditioning the exterior
FK bonds induces a wiring on the Q part that dominates the free boundary of
the box-alone disagreement-bond system, and the free FK law grows with the
set. Do the bond comparison, then the set comparison; average over the
exterior set. []
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PART 5. ORDERED TRANSMISSION ACROSS A CUT (5B: mass and the proof)
--------------------------------------------------------------------------------
For the rest of PART 5 the underlying graph in each local chart is the
critical square lattice (K_e = beta_c); integer roundings of rectangles
and segments are understood, and every lower bound holds once the relevant
scale is large (threshold depending only on the fixed geometry). Auxiliary
bonds below belong to the set sampled BEFORE the recolouring of Lemma 5.3;
so equal plus boundary data can still yield auxiliary disagreement-bond
connections even though the recoloured replicas agree.
Lemma tr:many-sites (many sites joined across a rectangle). Take two
independent plus-boundary Ising configurations in a rectangle of scale w
and fixed aspect ratio, and their auxiliary disagreement bonds. Let
L_1, L_2 be straight lattice segments of lengths comparable to w in fixed
compact bulk subsets, mutually separated by a distance comparable to w.
There are constants c, p > 0 (geometry only) such that
P_Q^{++}( |{x in L_1 : x <-> L_2}| >= c w^{7/8} ) >= p. (5.16)
Proof. In the independent original pair put Y_x = (eta_x - xi_x)/2.
There are no disagreement pins. Conditional cluster signs give
P_Q^{++}(x <-> y) = E[Y_x Y_y] = (1/2) Cov^+_Q(sigma_x, sigma_y), (5.17)
and more generally
P_Q^{++}(x <-> y, x' <-> y') <= E[Y_x Y_y Y_{x'} Y_{y'}]. (5.18)
Indeed if both pairs connect, every auxiliary cluster holds an even
number of the four indexed factors, so the conditional sign average is the
even-parity indicator (with all factors required to lie in the auxiliary
set); this also covers repeated indices. Put
X = sum_{x in L_1} sum_{y in L_2} 1{x <-> y}.
The bulk lower bound in Lemma 4.1(2) gives E X >= c_1 w^{7/4}. By (5.18),
Cauchy-Schwarz, and the fourth line moments of Lemma 4.3,
E X^2 <= E[ (sum_{L_1} Y_x)^2 (sum_{L_2} Y_y)^2 ] <= C_1 w^{7/2}.
(The Y-moments follow from either replica's moments by the triangle
inequality in L^4.) Paley-Zygmund gives c_2, p_2 > 0 with
P(X >= c_2 w^{7/4}) >= p_2.
Let L be the largest number of sites of L_2 lying in one auxiliary
cluster. Then L^2 <= sum over clusters C of |C\cap L_2|^2 and
E sum_C |C\cap L_2|^2 = E(sum_{L_2} Y_x)^2 <= C_2 w^{7/4}. Choose K with
C_2/K^2 < p_2/2; then with probability at least p_2/2 both
X >= c_2 w^{7/4} and L <= K w^{7/8}. Every site of L_1 reaching L_2
contributes at most L to X, so on that event at least (c_2/K) w^{7/8} sites
of L_1 reach L_2. The event in (5.16) is increasing (the event just used
was only for estimating its probability). []
Lemma tr:circuit (a surrounding disagreement circuit). In the doubled
plus-boundary model on B_{16w}, a circuit of actual disagreement bonds in
B_{4w} minus B_{3w/2} surrounding B_{3w/2} has probability at least a
constant p_0 > 0 for all large w.
Proof. Consider the four strips of the annulus, e.g. the top strip
[-4w,4w] x [2w,4w], extended in the long coordinate to [-5w,5w] with
agreeing plus boundary. Apply Lemma 5.5 (many-sites) to two vertical
segments at x = -4w and x = 4w with y in [5w/2, 7w/2]; they are in the
bulk. The connection stays inside the extended rectangle (the auxiliary
disagreement set cannot contain agreeing boundary pins), and a subpath is
a left-to-right crossing of the unextended strip. Local domination
(Proposition tr:local-domination) transfers the positive bound to
the doubled model on B_{16w}. Do all four strips. The four crossing events
are increasing so their intersection has probability at least the product;
on it, adjacent crossings meet inside the corner overlaps, and joining
successive intersection points gives a closed walk in the annulus with
winding one about the centre. Erasing closed subwalks leaves a simple
circuit of nonzero winding, all its edges in the four strips. []
Proposition tr:mass (mass at a source-connected patch). Fix a > 0. Let
b_0 be inside z + B_w and L inside z + B_w be a horizontal or vertical
lattice segment of length in [a w, 2w]. Suppose z + B_{16w} is a free
square-lattice neighbourhood (no common or disagreement pins inside). Then
in the doubled model with arbitrary ordered pins OUTSIDE this square,
sum_{x in L} P(x <-> U) >= c_a w^{7/8} P(U <-> b_0). (5.19)
(The constant and threshold depend only on a; the statement holds in an
isometric planar chart of a torus or cylinder.)
Proof. Translate z = 0. Take L' to be the translate of L by 6w in a
perpendicular coordinate. Both L and L' lie in fixed bulk subsets of
B_{16w}, and L' lies outside B_{4w}. Under the doubled plus-boundary law
on B_{16w}, Lemma 5.5 gives positive probability that at least
c w^{7/8} sites of L reach L' by paths internal to B_{16w}. Intersect this
increasing local event with the increasing circuit event of Lemma tr:circuit;
association gives their intersection probability at least a constant p > 0.
If P(U <-> b_0) = 0 there is nothing to prove. Otherwise condition on that
increasing event and apply local domination (tr:local-domination): the
circuit and many-sites events still hold jointly with conditional
probability at least p. Every path from U (outside B_{16w}) to b_0 (inside
B_w) meets the surrounding circuit, and every counted path from L (inside
B_w) to L' also meets it; by planarity all counted sites are hence
connected to U. The conditional expected number of counted sites is at
least p c w^{7/8}; multiplying by the conditioning probability proves
(5.19). []
5.6 Proof of Proposition tr:transmission. Use the monotone coupling of
Lemma 5.3: let eta_J >= xi_J be its cut configurations. Put
k(j) = E[h | sigma_J = j], the conditional expectation using the COMMON
target-side continuation. By the spatial Markov property and the
separating property, k does not depend on which source configuration was
prescribed, so
Delta h = E[ k(eta_J) - k(xi_J) ].
In this telescope the other cut spins are common pins for the pointwise
comparison; the local mass estimate (tr:mass) is applied to the original
coupling, whose counting charts stay pin-free.
Interpolate between eta_J and xi_J one patch at a time in a deterministic
order. Intermediate configurations are legitimate cut configurations, the
other patches fixed. Corollary 3.5 bounds the oscillation of k in one patch
by C||h||_inf (w/s)^delta (with these arbitrary other cut spins as common
pins). If the two sampled cut configurations agree on the patch the
summand is zero; by (5.5) they disagree somewhere in b_i exactly when
U <-> b_i. Hence
| Delta h | <= C ||h||_inf (w/s)^delta sum_i P(U <-> b_i). (5.20)
Proposition tr:mass applied at scale ell_i ~ w, plus the bounded overlap
of counting segments, gives
sum_i P(U <-> b_i)
<= C w^{-7/8} sum_i sum_{x in L_i} P(U <-> x)
<= C K w^{-7/8} sum_{x in J} P(U <-> x).
The last sum equals (1/2) s^{7/8} Delta M_J by (5.5) (summing the identity
E eta_x - E xi_x = 2 P(x <-> U) over x in J and using (5.2)). Substituting
in (5.20) gives (5.3). All local constructions use the same fixed aspect
ratios and overlap constants, independent of w/s. []
5.7 Signed and random sources. Definition tr:order-positive. For a law mu
on the source spins, an ORDER-POSITIVE CENTERED DENSITY is a real q with
E_mu q = 0 and E_mu[q f] >= 0 for every increasing source function f.
(Here "density" means a signed measure; q need not be nonnegative or
increasing.)
Corollary tr:signed-transmission. The conclusions of Proposition
tr:transmission hold after averaging over any ordered coupling of two
random source configurations, with the appropriate common Gibbs
continuation. In particular, for a Gibbs law mu with common pins and an
order-positive centered density q = q(sigma_S) of its source marginal,
| E_mu[q h] | <= C ||h||_inf (w/s)^{-7/8+delta} E_mu[q M_J]. (5.21)
The constants do not depend on the mass of the positive part of q.
Proof. For an ordered random pair of sources, apply (5.3) to each pair and
integrate; magnetic differences are nonnegative, so the triangle inequality
bounds the integral of the target difference by the integral of its
absolute value. For the signed form, put m = E_mu q_+ = E_mu q_- (equal by
E q = 0); if m = 0 there is nothing to prove. Otherwise
mu_+- = m^{-1} q_+- mu_S are probability laws and order positivity says
mu_+(F) >= mu_-(F) for every increasing F, so they admit a coupling
(s_+, s_-) with s_+ >= s_-. (Finite transport: send mass mu_-(x) from each
lower configuration to upper ones with capacities mu_+(y); the needed
cut inequality mu_-(B) <= mu_-(up(B)) <= mu_+(up(B)) is exactly what order
positivity provides.) Apply the random-source assertion to this coupling
and multiply by m; the Gibbs continuation turns the difference of source
integrals into E_mu[q h] and E_mu[q M_J]. []
Corollary tr:ordered-tv (ordered changes of domains and total variation).
Suppose two ordered Ising laws (possibly different pin sets or volumes)
have a common free source contour S_0 from which their conditional
continuation toward a target T is the SAME, and let an admissible cut J lie
between S_0 and T. Then (5.3) holds for their expectation differences, and
in particular, with total variation = sup over event probability
differences,
|| mu_T^+ - mu_T^- ||_TV
<= C (w/s)^{-7/8+delta}
( E_{mu^+} M_J - E_{mu^-} M_J ). (5.22)
At fixed positive w/s, convergence of the magnetic difference to zero
forces convergence in total variation on the buffered target.
Proof. Restrict an ordered coupling to S_0 and use the common Gibbs
continuation. Apply tr:signed-transmission in its random-source form, taking
h to be an arbitrary target-event indicator for the TV supremum. All
differing pins must lie on the source side of S_0; the common-continuation
hypothesis excludes a changed pin in the target component. []
Interpretation. (5.22) is the conversion rule that powers PART 9: if we can
sandwich a real (irregular) target law between two regular-comparison laws
and make the MAGNETIC difference on a cut small, then the TARGET total
variation is small. The exponent 7/8 - delta is the price of a narrow
(source-band) collar relative to the distance s to the target.
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
PART 6. NEARBY CONTOURS AND PREDICTION
--------------------------------------------------------------------------------
This is a separate consequence of the correlation inputs, independent of
the stopping construction. It shows that one normalized magnetization can
be predicted from a nearby contour. The relative displacement h = w/s is
held fixed while s -> infinity; the proof takes the continuum limit of the
two-point sums first, then estimates the change under displacement, so no
discrete derivative of the two-point function is needed.
6.1 Setup. Recall M_{J_s} = s^{-7/8} sum_{x in J_s} sigma_x. Let
J_s, J'_s be two parallel horizontal or vertical lattice segments of
lengths of order s whose MATCHING portions differ by a displacement of
length at most C w, and whose unmatched end pieces have total length O(w).
The same will be allowed for two concentric square contours of radii
comparable to s and radius difference at most C w, by matching their four
sides. Assume the union has a free neighbourhood of width c s and is
covered by a fixed number of buffered planar charts. Write
M_{J_s} = s^{-7/8} sum_{J_s} sigma_x and M_{J'_s} = s^{-7/8} sum_{J'_s}
sigma_x.
Proposition tr:line-difference. There is a constant C_0 (fixed geometry
and buffer) such that for every fixed h = w/s in (0, h_0],
limsup_{s -> oo} || M_{J_s} - M_{J'_s} ||_2
<= C_0 h^{3/8}. (6.1)
The law may have arbitrary common pins beyond the buffer or be a mixture
of such laws. The same holds for nearby full periodic rows of period W
(normalized by s^{-7/8}) when W/s is fixed and positive, with constant
depending on W/s.
Proof. The buffered comparison (Corollary 3.4) applied to the
NONNEGATIVE SQUARE of a line difference reduces each buffered chart to the
whole-plane law at the cost of a fixed factor; a finite chart decomposition
and the triangle inequality in L^2 then suffice.
In the whole-plane law the mean is zero, and by Lemma 4.1(3) normalized
two-point sums on rescaled segments converge to the corresponding integrals
of
K(x-y) = c_sigma |x-y|^{-1/4}.
If the endpoint data vary with s, pass to a subsequence realizing the
limsup, then translate, then to a further subsequence on which rescaled
lengths and relative endpoints converge; these data range over a compact
set and all estimates below are uniform on it.
Diagonal control. The uniform bound
E(sigma_x sigma_y) <= C (1 or |x-y|)^{-1/4} shows that parameter pairs at
rescaled distance at most r contribute at most C r^{3/4} + o(1) to the
second moment of a normalized segment sum. Equal-site terms are
O(s^{-3/4}). Off the diagonal the two-point asymptotic is uniform, so
ordinary Riemann sums converge. This justifies the integral limit for self
and cross terms, including shifted segments.
Matching segments, near and far. Pair the points of the two matching
segments by a displacement v with |v| <= C h. In the limiting squared
difference, split pairs of the one-dimensional parameters at distance
2 C h.
- NEAR part (parameters within 2 C h): bounded by a constant times
integral_0^{C' h} r^{-1/4} dr = O(h^{3/4}); the same bound applies if a
shifted kernel has its singularity at a nearby parameter value.
- FAR part: |grad K(u)| <= C |u|^{-5/4}; the mean value theorem bounds the
combined kernel difference by C h |u|^{-5/4}, whose integral is at most
C h integral_{2 C h}^{C'} r^{-5/4} dr = O(h^{3/4}).
So the limiting squared norm of the matching-segment difference is
O(h^{3/4}), i.e. O(h^{3/8}) in norm.
Unmatched ends. An unmatched end of length O(h) has second moment
O(h^{7/4}) by the same kernel integral (integral over r in [0,h] of
r * r^{-1/4} dr = O(h^{7/4})), hence contributes O(h^{7/8}) in norm. There
are only boundedly many such pieces, and h^{7/8} <= h^{3/8} for h <= 1, so
the triangle inequality gives (6.1).
Concentric squares and periodic rows. For concentric squares, trim
corresponding sides to equal lengths, translate the matching portions in
the perpendicular direction, and apply the segment result to each side and
to the short remainders. For full periodic rows, subdivide both rows at the
same fixed locations into matching segments of length a small fixed
fraction of W; each pair with its buffer lies in a planar chart (decrease
h_0 if necessary by a constant depending on W/s), apply the comparison, and
sum the finitely many segment norms. Mixtures are permitted because the
density comparison is uniform before mixing. []
Corollary tr:line-regularity (prediction from a nearby contour). In the
setting of Proposition 6.1, let F be a sigma-field containing the spins of
J'_s, and write P_F X = E[X | F]. Then
limsup_{s -> oo} || M_{J_s} - P_F M_{J_s} ||_2
<= C_0 (w/s)^{3/8}. (6.2)
The normalization on the approximating sum stays s^{-7/8} even when the two
contours have different lengths.
Proof. M_{J'_s} is F-measurable, and the conditional expectation minimizes
the L^2 distance among F-measurable functions:
|| M_{J_s} - P_F M_{J_s} ||_2 <= || M_{J_s} - M_{J'_s} ||_2. Apply (6.1). []
Remark tr:quantifiers (order of scales). The transmission exponent delta,
the aspect ratios in the local mass construction, and the constants for a
fixed family of cuts are chosen FIRST. The bounds are uniform for
sufficiently large patch scales. For each of N concentric cuts at spacings
comparable to s/N, the local geometry is the same for every N and
w/s ~ 1/N, so the constant in the transmission lower bound is independent
of N. For the nearby-line estimate, fix N and then take the mesh
sufficiently fine (this threshold may depend on N). A large fixed N can
therefore be selected using the transmission and line constants BEFORE
taking the mesh limit.
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
PART 7. COMPLETE INTERFACE RESOLUTIONS
--------------------------------------------------------------------------------
The stopping construction (PART 8) uses a FIXED face triangulation, whereas
the published loop limit uses LEFTMOST interfaces. We must compare COMPLETE
collections, including the number and orientation of their macroscopic
loops: changing one face pairing can merge two loops or split one. This
part proves that every complete resolution has the same CLE_3 limit.
7.1 Circuits. Use the oriented-loop convention of [Benoist-Hongler 2019,
arXiv v3, Section 2.5]: an Ising CIRCUIT is a closed dual path, with no
repeated edge, having plus spins on its left and minus spins on its right.
It may visit a dual vertex twice. At a CHECKERBOARD FACE (a square whose
four corner spins alternate) there are exactly two noncrossing ways to
pair the two incoming and the two outgoing disagreement edges. A COMPLETE
RESOLUTION chooses one pairing at every checkerboard face and uses every
disagreement edge exactly once. Fixing a diagonal in each square face and
taking the zero set of the affine spin interpolation on its two triangles
gives one such resolution, up to a displacement of order delta.
A loop is LEFTMOST when the adjacent plus spins along its left side are
connected successively by nearest-neighbour steps (the strong-path
convention of [BH2019, Section 2.5]). The canonical leftmost collection is
all leftmost loops.
7.2 Distance between loop collections. For oriented curves let d_u be the
uniform distance modulo orientation-preserving parametrization; for loops
also allow a change of starting point. Distances between collections are
obtained by matching loops and charging the diameters of unmatched loops,
as in [BH2019, Section 2.7].
Lemma if:resolution-convergence. In a plus Jordan domain whose lattice
boundary converges uniformly modulo parametrization, EVERY complete
resolution of the spin disagreement edges converges to the same oriented
nested CLE_3 collection. The assertion is uniform over the choice of the
resolution; in particular it applies to a fixed choice of diagonal in every
square face.
Proof. Brick C4 = [BH2019, Theorem 6]: the leftmost loops converge in the
collection metric, AND the following SIMULTANEOUS APPROXIMATION holds. For
every e > 0, with probability tending to one as delta -> 0, every Ising
circuit l of diameter greater than e has a leftmost circuit l^L with
d_u(l, l^L) <= e,
diam C < e for every component C of (l union l^L) minus (l intersect l^L).
The paragraph after that theorem adds: shared edges become dense in each
macroscopic loop. In particular, above a fixed positive diameter cutoff
every circuit shares edges with its matched leftmost circuit, w.p. -> 1,
for ALL circuits simultaneously on the same event (including circuits
selected after seeing the spin configuration).
Coverage and multiplicity. From any vanishing mesh sequence, choose errors
e -> 0 slowly enough that these simultaneous events have probabilities
-> 1. Pass to a subsequence and couple the leftmost collections to their
limit, so that both the collection-distance error and the all-circuits
error tend to zero almost surely. Choose a diameter cutoff that is NOT the
diameter of a limit loop. Only finitely many limit loops exceed it; they
are simple, pairwise disjoint, and distinct members have positive mutual
distance.
Any two complete resolutions are joined by changing checkerboard pairings
ONE AT A TIME. Both pairings match incoming to outgoing oriented edges;
each change MERGES two edge-simple circuits or SPLITS one into two. So every
intermediate circuit stays in the class covered by the all-circuits
assertion (no crossing circuit, no repeated edge is introduced).
Suppose, along finer meshes, that some such switch changes the number of
circuits matched near a fixed positive-diameter limit loop. Select one
offending switch per mesh (the all-circuits assertion holds even for
configurational-dependent selection). After a subsequence, each constituent
either has vanishing diameter or converges modulo parametrization to a
simple loop of the limiting collection. Two vanishing constituents cannot
merge to macroscopic size (they meet at the switched vertex). Adding a
vanishing constituent to a large one does not change the latter's
parametrized limit. If both constituents are large, their meeting vertices
force their limit loops to meet, so the limits coincide.
The last option is impossible. Pick a point strictly inside that simple
loop; uniform parametrized convergence preserves its winding number. Each
constituent has winding 1 or -1, whereas the merged traversal has their sum
0 or +-2. The merged circuit is itself covered by the simultaneous
assertion, its image approaches the same loop, so uniqueness of the nearby
limiting loop makes its oriented parametrized limit that loop too, with
winding +-1, a contradiction. Splits are the reverse. Hence NO switch can
change the macroscopic count for all sufficiently fine meshes on this
coupled subsequence (the argument picks an offending switch if one exists,
so it does not accumulate error over the number of switches).
A specified leftmost circuit CAN be included in a complete resolution: fix
its pairings at every visited vertex and resolve the unused edges
arbitrarily (at a once-visited checkerboard vertex the unused incoming and
outgoing edges have the remaining noncrossing pairing; at a twice-visited
vertex the circuit already fixes both pairs). In this resolution a second
circuit matched to the same limit loop would have to share edges with the
specified leftmost circuit (uniqueness of the nearby macroscopic leftmost
loop plus the common-edge consequence), contradicting use of every edge
once. So the count is one here, and the switching argument makes it one in
EVERY resolution. The all-circuits assertion also excludes unmatched
positive-diameter circuits. This proves convergence in the collection
metric.
Colors are recovered by NESTING from exterior plus. An ancestor of a loop
of diameter at least a also has diameter at least a, so only finitely many
ancestors exceed any positive scale; their nesting and orientations are
stable under convergence to disjoint simple loops, so colors at positive
scales converge too. []
7.3 Why this matters for the stopping picture. Any resolution mode we
pick (e.g. the fixed face diagonals) yields the SAME macroscopic nested
CLE_3 boundary picture. Therefore when we later stop right ends at a level
set, the resulting visible boundary is built from limit loops that are
simple, disjoint, locally finite, and independent of the mesh resolution
mode, which is exactly what PART 8 needs to obtain a Jordan base with
finitely many slits.
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
PART 8. STOPPED INTERFACES AND SIGNED SLIT DATA (8A: geometry)
--------------------------------------------------------------------------------
We now convert the complete loop limit into the boundary information that a
conditional target law actually needs. The exploration keeps its full spin
transcript, while the geometric DATUM records only the boundary VISIBLE
from the target. PART 9 replaces that boundary uniformly over nearby
transcripts. All distances here and in PART 9-10 are macroscopic: the
lattice is delta Z^2, with delta -> 0.
8.1 Band geometry. Write Q_r = { z in R^2 : ||z||_inf < r } and
gamma_r = boundary of Q_r. Fix 0 < u < v and positive clearances from the
band closure(Q_v) minus Q_u to the target and to the boundary of the
ambient chart. There are two versions.
(INWARD) reveal the spins OUTSIDE Q_v first, and explore from gamma_v
toward gamma_u. The target is a fixed compact subset of Q_u (in
applications, a strictly smaller concentric square).
(OUTWARD) reveal the spins INSIDE Q_u first, and explore toward gamma_v.
There is a more distant fixed square wall, and the target lies between
gamma_v and that wall, with positive clearances from both. The distant
wall has prescribed signs (including a constant minus wall).
The statement also allows a compact INTERVAL of choices for each radius,
provided the intervals are separated and all stated clearances hold
uniformly. For a requested radius r, choose deterministically the nearest
r_delta in delta Z, ties by a fixed rule, and use the contour
partial Q_{r_delta}. For small delta these rounded radii preserve all
prescribed separations, and their contours follow primal grid edges. The
interpolated zero set meets such a contour ONLY at edge midpoints: it never
passes through a spin vertex, and the two adjacent triangles lie on
opposite sides of the contour edge. So these meetings are crossings,
including near the square corners. In either version, include the vertices
of the rounded starting contour in the initial revealed layer. For
measurability, the BAND SPINS are the vertices of all triangles meeting the
closed rounded band; this adds only an O(delta) layer and includes the
starting-contour vertices.
8.2 Remark if:thin-collars. No universal lower bound on v/u - 1 is
required. Each collar has strictly positive width fixed before the mesh
limit, and the constants/threshold may depend on that width. Example:
collinear segments of length ell, separated by gaps g > 0, fit in disjoint
squares of halfside ell/2 + g/8 and far-wall halfside ell/2 + g/4, leaving
room between them for any fixed finite list of nested contour levels with
positive clearances. For pieces from a polygonal contour, remove
neighbourhoods of corners and boundary intersections first, then take
segments shorter than the separations; the straight pieces admit the same
construction. Constants may depend on g/ell; no fixed enlargement factor
independent of g/ell is needed.
When radii vary, reserve one fixed larger permitted WINDOW containing their
compact ranges, and choose all common source and test contours with
positive clearance from the whole range. The regular barriers built below
stay on the fresh side of the realized visible wall with the prescribed
target clearance; they need NOT lie between each realization's numerical
radii u, v. The reserved window provides room for their offsets and caps.
8.3 Recording map R_delta. We encode lattice boundary data on the REQUESTED
continuum contours. Choose disjoint fixed intervals about the two permitted
radius ranges, with positive endpoint clearances. Let f_delta be the
increasing piecewise-linear map of [0,oo) that sends u_delta to u and
v_delta to v, fixes the interval endpoints, and is the identity outside
them, linear between consecutive knots. Define
R_delta(0) = 0, R_delta(z) = ( f_delta(||z||_inf) / ||z||_inf ) z
for z != 0. (8.1)
This homeomorphism and its inverse move every point by O(delta),
uniformly over permitted radii (on each affine interval, f_delta(r) - r
interpolates endpoint errors of size at most delta/2). It maps the closed
rounded band onto the exact band and each rounded contour onto its
requested counterpart. We apply R_delta ONLY when recording the geometry
of the exploration; the physical spins, reveal, and Gibbs laws stay on the
original lattice.
8.4 Reference law. For the probability limit, sample spins in a fixed
larger square with PLUS boundary conditions. Other buffered laws enter only
through Corollary 3.4. The sign-reversed reference square may equally be
used. A finite collection of band observations is always computed from ONE
common spin configuration, so its joint law is unambiguous.
8.5 The exploration. Fix ONE diagonal in every square face, without
adding interactions to the Ising Hamiltonian. Affinely interpolate the
spins in each triangle. The zero set consists of noncrossing polygonal
strands. After the initial reveal, INSPECT every strand meeting the
starting contour. Follow it on the fresh side until its FIRST RETURN to the
starting contour or its FIRST HIT of the other contour. In the latter case
also follow the strand from its other starting endpoint, when there is one,
by the same rule. Every time a triangle is traversed, inspect its vertex
spins. The order of exploring the starting endpoints is fixed
deterministically. Repeated inspections reveal nothing new.
From the target, the VISIBLE boundary is a base formed by the starting
contour and returning excursions, together with strands STOPPED at the
second contour and their two side signs. This is the target-facing datum of
Definition 8.7.
Lemma if:stopping. For each delta > 0 this is a FINITE adaptive spin
reveal. Conditional on its complete information, the unexamined spins have
the Ising Gibbs law with the examined spins fixed. The revealed set lies on
the starting side and in an O(delta) enlargement of the band. The traced
strands, their side signs, and the target-facing boundary datum can be
computed from the BAND SPINS alone.
Proof. There are finitely many starting edges and finitely many triangles.
At each step the next triangle is determined by the previously exposed zero
segment and the fixed diagonal convention; its spins determine the
continuation. So the event that a specified reveal transcript occurs
depends only on the spins recorded in that transcript. In the finite Gibbs
density, conditioning on such a transcript fixes exactly those spins and
leaves the original interaction weights on all remaining variables. This is
the Gibbs (stopping) property directly, including a random examined set.
The trace stops at the second contour and examines only adjacent triangles,
proving the location claim. Although the initial reveal contains a whole
side of the band, the algorithm deciding a strand within the band uses only
the spins adjacent to the starting contour and in the band; the
target-facing datum is determined by these traces and their local signs. No
spins in the rest of the initially revealed side are needed. []
--------------------------------------------------------------------------------
8.6 The visible Jordan base and its slits. These definitions refer first
to an oriented collection of simple disjoint continuum loops. Restrict a
loop to the fresh side of the starting contour. A maximal such portion
that returns to the starting contour before reaching the second contour is
a RETURNING CHORD. A portion that reaches the second contour is retained
only up to its first hit there and is a STOPPED STRAND. Cut the fresh
region along these curves, and take the component containing the untraced
region beyond the second contour; retain only boundary pieces seen from
this component. The same definition is made for resolved lattice walls.
The boundary is read from the remaining domain: a traversal follows one
side of each slit to its tip and back along the other side. So the two
sides of a slit are DISTINCT boundary arcs even though they share a
geometric image. This is the PRIME-BOUNDARY traversal used below.
Definition if:datum (admissible facing datum). A facing datum consists of
an ORIENTED Jordan base, finitely many pairwise disjoint simple SLITS
attached to it at distinct ROOTS, the root and TIP marks, and signs on the
two sides of each slit and on the intervening base arcs. Signs on a slit
are OPPOSITE. In the prime-boundary traversal the sign changes at a TIP and
stays unchanged through each occurrence of its ROOT. The whole datum lies
in the closed band, and its tips lie on the second contour. With NO slits,
the base has one constant side sign.
The LATTICE datum is the image of its physical boundary record under
R_delta, retaining roots, tips, orientation, and side signs. When radii
vary over permitted windows, X_band denotes the union of admissible data
from those radius choices; the numerical radii are NOT extra marks (e.g. a
datum with no slits need not determine its second radius).
Metric on X_band. Within a fixed finite combinatorial type, use the MAXIMUM
uniform distance modulo increasing parametrizations of the base and slits,
respecting root attachments, side labels, and orientation. Cap the distance
at one, and set distance one between different types.
Ambient space X^. We also need centres for approximation: take all finite
marked CONTINUOUS walks of these combinatorial types, keeping
orientations, side labels, root and tip marks, and the shared coordinates
of attached roots, but DROPPING simplicity, disjointness, and the exact
band-containment and contour-incidence requirements. Use the same
matching-distance formula, identifying representatives at distance zero.
Then X_band sits inside X^ with X_band's metric the induced one. A ball
with an ambient prototype as centre always means its intersection with
X_band when used as a set of facing data.
Lemma if:visible-data. For almost every choice of the two contour radii,
the continuum construction from nested CLE_3 yields an admissible facing
datum. Its base may contain infinitely many returning chords but IS a
Jordan curve. The lattice data converge to it in X_band. These assertions
hold jointly for any fixed finite collection of bands.
Proof. Work on the probability-one event that the loops are simple and
disjoint and locally finite at every positive diameter, and prove the
statement deterministically there.
Generic levels. For a continuous real function on a compact parameter
interval, its local maximum and minimum values form a COUNTABLE set (each
is the max/min on an interval with rational endpoints inside a witnessing
neighbourhood). Apply this to the square radius along each loop. There are
countably many loops, so the union of exceptional levels is countable for
each collection, and Fubini gives a deterministic full-measure set of
radius choices at which there is no one-sided contact and no constant
interval.
Finiteness at scale a. Only finitely many excursions have diameter greater
than any prescribed a > 0: only finitely many loops have that diameter,
and on each, uniform continuity of a parametrization gives a positive
minimum parameter length for such an excursion; these parameter intervals
are disjoint. A stopped strand has diameter at least the positive distance
between the two contours, so there are finitely many stopped strands, and
their tips are distinct (the two stopped ends of one deep excursion could
share a tip only at a one-sided contact at the second level, which was
excluded).
Distinct endpoints. Distinct excursion endpoints on the starting contour
are distinct: endpoints on different loops cannot meet by disjointness, and
a shared endpoint for two excursions of one simple loop would be a
one-sided level contact. Excursion interiors miss the starting contour, and
simplicity and disjointness exclude all other intersections.
Deletion intervals. Each visible returning chord deletes the open interval
of the starting contour cut off from the target component. Visible deletion
intervals have disjoint interiors (in a pair of nested deletions the inner
chord is screened). A SMALL chord deletes a SMALL interval: join its
endpoints by the short arc of the starting square; as the chord diameter
tends to zero, uniform continuity of the inverse parametrization of that
square makes the arc small, so the resulting small Jordan pocket cannot
contain the target beyond the second contour (a fixed positive distance
away); the erased interval must be that short arc, not the complementary
long one. (Works also on the exterior side; compactify by a fixed inversion
away from the band.)
Jordan parametrization. Parametrize the starting contour and replace each
deleted interval by a parametrization of its chord with the same endpoints,
keeping the original parametrization elsewhere. At an accumulation of
replacement intervals, the interval diameters and chord diameters tend to
zero (by the finiteness and small-pocket facts), so the resulting map is
continuous; it is injective (chord interiors miss the starting contour and
one another, endpoints are distinct, retained points stay distinct). A
continuous injection of the compact circle into the plane is a Jordan
parametrization. So infinitely many small returning chords do NOT create
an accumulating pinch. Adding the finitely many visible stopped strands
gives the asserted base and slits.
Continuity in X_band. Choose a > 0 outside the countable set of excursion
diameters and retain the finitely many excursions of diameter greater than
a. Uniform convergence of the parent loops gives convergence of these
excursions, their endpoints, and their first hits of the second contour.
The generic levels enter here: a compact subarc strictly on one side of a
level stays there under a small uniform perturbation, while arbitrarily
near an excursion endpoint on the other side in parameter time the
limiting loop has points on the opposite side of the level (else it would
be a one-sided contact). These points BRACKET the perturbed endpoint
between arbitrarily nearby parameter values; the same at the cutoff
brackets its first hit. Splitting, merging, or losing a retained excursion
would contradict a bracketing property.
The circular order of the finitely many retained endpoints is stable, and
so is their visibility: an additional chord of small diameter cannot
screen a retained chord, because its deleted pocket has small diameter and
cannot contain that retained chord. After matching the retained pieces,
match the remaining base intervals in circular order; all discrepancies on
them have diameter tending to zero as a -> 0. This builds parametrizations
of uniformly small distance. Include the finitely many slit roots; their
stopped strands converge by the same bracketing. The fixed band width and
the generic second level give eventual equality of the finite stopped-
strand list, with its cyclic root and tip order. Only the returning chords
need the subsequent a -> 0 limit. Finally R_delta adds O(delta) and
ensures the lattice records have the exact contour incidences required by
the data space.
Signs. At the lattice level a returning chord replaces an interval whose
two surviving side signs AGREE, so the only sign changes in a traversal of
the visible prime boundary occur at stopped tips: microscopic returning
chords introduce no extra alternation between tips. When slits are present,
their oriented sides determine the signs on all intervening base arcs, so
those signs converge by Lemma 7.2 (if:resolution-convergence).
No-slit constant sign. To recover the constant sign when there are no
tips, use one further property of nested CLE_3: a deterministic interior
point is almost surely surrounded by infinitely many nested loops [SSW
2009, Section 1, before Proposition 1]. Their diameters tend to zero by
local finiteness. Choose such a point on the starting contour. A
sufficiently small surrounding loop misses the second contour and has a
returning excursion into the fresh side. If this chord is hidden, any
screening chord has diameter bounded away from zero by the small-pocket
argument; there are only finitely many such chords, so one is visible. Its
oriented parent loop supplies the constant side sign, and that loop with
its finitely many ancestors converges WITH ITS ORIENTATION by Lemma 7.2.
This additional fixed-point nesting is used ONLY for this colour anchor,
not for the deterministic construction of the Jordan base.
The deterministic argument applies simultaneously to any fixed finite
list. Lemma 7.2 and the continuous mapping theorem finish. []
Remark if:metric-separable. The ambient space X^ is separable: approximate
finitely many parametrized walks by rational piecewise-linear walks,
approximating shared endpoints and roots together. These prototypes need
not be simple or satisfy the exact band constraints; they form a countable
dense set in the ambient matching metric. The countable union over finite
combinatorial types stays separable, and an ambient countable base
restricts to a countable base on X_band, so that subspace is separable too.
Only REALIZABLE admissible data, not the prototypes, are used to construct
Gibbs boundaries.
--------------------------------------------------------------------------------
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PART 9. REGULAR BOUNDARIES AND ORDERED EROSION (9A: chart and convergence)
--------------------------------------------------------------------------------
The visible boundary from PART 8 is a Jordan base with finitely many
signed slits. We now replace it by ONE deterministic regular boundary,
uniformly over every sufficiently near realizable stopping transcript (whose
examined sets may differ). This uniform local statement feeds the finite
probability cover of PART 10.
9.1 Reduction to spin means. Choose a source contour S_0 and a square test
contour J in the FREE region between the entire permitted stopping band and
the target, with positive collars chosen so J is an admissible cut between
S_0 and the target (Definition 5.1). All modifications of the facing
boundary occur on the BAND side of S_0. So for each fixed distant-wall
assignment the two laws in an ordered insertion share the SAME conditional
continuation from S_0 toward the target. Write (matching (5.2) with line
scale)
M_J^delta = delta^{7/8} sum_{x in J^delta} sigma_x. (9.1)
Lemma 4.3 bounds its second moment uniformly. The fixed macroscopic
reference scale only changes the constant in Corollary 5.x
(tr:ordered-tv).
Lemma if:erosion (local ordered erosion). Fix an admissible facing datum d,
the band geometry, a target beyond the band with a free collar, and
eps > 0. There are a neighbourhood U_d of d in X_band, a regular Jordan
contour Gamma_d on the FRESH side of d with finitely many signed arcs, and
delta_d > 0, such that for every 0 < delta < delta_d and every realizable
stopping transcript whose datum lies in U_d:
the lattice boundary layer following Gamma_d is FREE in the original
conditional law. With mu^0 that law, mu^+ obtained by inserting Gamma_d's
plus pins, and mu^f by then inserting its minus pins, the two changes are
ordered and
|| mu^0|_T - mu^+|_T ||_TV + || mu^+|_T - mu^f|_T ||_TV < eps. (9.2)
On the fresh side of the full contour, mu^f is EXACTLY the Gibbs law
determined by this contour (and, in the outward case, the unchanged distant
wall): it depends on the contour, the mesh, and the distant wall, NOT on the
transcript within U_d. The contour can be polygonal or smooth with switches
on smooth portions; its clearances, neighbourhood, and mesh threshold may
depend on d and eps but not on remaining spins or the transcript.
What the construction has to show. Take any candidate barrier on the band
side of S_0 whose layer is free in the original law, and the three laws
above. Put m_j = E_{mu^j} M_J^delta for j in {0,+,f}. Inserting plus pins is
UPWARD and inserting minus pins afterward is DOWNWARD, so m_+ >= m_0, m_f.
Corollary tr:ordered-tv gives
(LHS of 9.2) <= C [ (m_+ - m_0) + (m_+ - m_f) ]. (9.3)
We build two deterministic regular signed boundaries (upper, lower)
approximating the same slit datum, with test-contour means m_u, m_l. After
changing pins only in small neighbourhoods of the finitely many roots and
tips, boundary monotonicity places every nearby transcript law BETWEEN these
two regular laws. If the resulting mean error is <= e, then
m_l - e <= m_j <= m_u + e (j in {0,+,f}),
and the right of (9.3) is at most 2C(|m_u - m_l| + 2e). The small-patch
comparison makes e small; marked-domain convergence makes the two regular
means approach the same limit. For an outward band the analysis first uses a
constant distant wall; Lemma fd:outer-wall-weight (PART 9B) restores
arbitrary prescribed wall signs at a uniform cost.
9.2 Regular outlines need a chart. An admissible datum bounds a disk with
finitely many attached slits (inward) or an annulus with slits attached to
its inner boundary (outward; outer wall unchanged). Opposite signs on the two
sides of a slit refer to its PRIME-BOUNDARY sides; actual Gibbs boundaries
will be narrow outlines with separate sides, not two signs on one vertex.
Lemma if:chart (a chart for a finite slit graph). Let a Jordan base have
finitely many disjoint attached simple arcs at distinct roots. Then on a
neighbourhood of this graph there is a homeomorphism making the base a
regular circular line and the arcs disjoint straight segments on its fresh
side, with a further open collar beyond every tip, and a fixed compact
target disjoint from the graph stays outside a sufficiently small such
neighbourhood.
Proof. Subdivide the base at its roots and add vertices until base plus arcs
form a finite simple plane graph. [Schaefer 2021, Theorem 1] (isomorphic =
homeomorphic by a plane homeomorphism) gives a polygonal embedding by an
ambient homeomorphism; the free tip of an attached arc has a neighbourhood
where it is a straight half-edge, prolongable a short way in the complement
(no common metric modulus). The Jordan-Schoenflies plane extension theorem
[Moise 1977, Section 10, Theorem 4] supplies a collar of the base.
Compatible prolongations: prolong each slit from its free tip to a
surrounding curve on the fresh side, disjointly, one at a time: an arc
attached at one endpoint does not separate the adjacent open region, so its
free end can reach the surrounding curve without crossing other arcs; cut
along the chosen arc and repeat; choose distinct endpoints in the cyclic
order of the roots. The prolonged arcs cut the collar into finitely many
Jordan cells; map their boundary arcs compatibly to the sides of rectangular
cells in a circular strip (each tip to an interior point of a straight
side), apply the plane extension theorem in each cell, and glue: the maps
and inverses are continuous (finitely many closed cells with agreeing
boundary maps). No slits: the initial collar suffices. The surrounding curve
and prolongations lie in a neighbourhood of the finite graph disjoint from
the target (small regular neighbourhood after extension); compactness and
positive distance give the clearance. []
Henceforth a displacement such as q or rho is measured in THIS chart; we do
NOT assume it is Lipschitz. Once finitely many chart parameters are fixed,
disjoint compact pieces have positive PHYSICAL separation, and the data
tolerance and mesh are chosen below it. Undoing R_delta moves the actual
wall by O(delta), so a small recorded-data tolerance followed by a fine mesh
gives any prescribed chart tolerance; all separation and incidence
statements below absorb this into positive slack.
Lemma if:regularization (small regularization of an outline). A Jordan
outline, finitely many marked points on it, and a sufficiently small
surrounding Jordan collar admit a polygonal or smooth replacement through an
ambient homeomorphism supported in that collar, with arbitrarily small
displacement; the marks keep their cyclic order and can be placed on
straight grid-aligned portions. Finitely many disjoint outlines can be done
at once.
Proof. Schoenflies-parametrize a closed annular collar; uniform continuity
lets it be thin and cyclically subdivided into small quadrilateral strips
with small open overlaps (nonconsecutive strips disjoint), marks away from
overlaps. Choose a joining point in each overlap, join successive points by
polygonal arcs inside the intervening open strip erasing loops, and bridge
across each overlap by truncating before the first intersection and adding a
simple polygonal bridge (local, overlaps disjoint). This gives a Jordan
polygon in the prescribed cyclic order with all endpoints in open sets (no
straight approach to a rough cell boundary needed); insert a short segment
at each marked strip. Transverse crosscuts split old and new collars into
corresponding Jordan cells (identity on the outer boundary, old arc mapped
to new arc, marks matched, agreeing maps on crosscuts); the plane extension
theorem extends cell by cell and glues to a homeomorphism equal to the
identity outside the collar, with displacement bounded by the thin strip
diameter (arbitrarily small) and the same bound for its inverse. Disjoint
collars allow simultaneous construction. []
Lemma if:marked-convergence (marked convergence of ribbon outlines). Let D
be a disk or annulus bounded on one side by an admissible slit datum and, in
the annular case, on the other side by a fixed disjoint Jordan wall with a
fixed finite collection of nondegenerate signed arcs. Replace each slit by a
ribbon of width and displacement -> 0 in the chart of Lemma 9.2, offset the
base by a quantity -> 0, use straight strips away from nodes and fixed
polygonal templates (no narrowing neck beyond a fixed fraction of their
scale) at the finitely many nodes, and regularize the resulting outlines by
physical displacement -> 0 (Lemma 9.3), preserving boundary-component labels
and cyclic order of side signs, with one cap mark per slit tip. Then the
domains D_q converge to D in the marked Caratheodory sense, normalized
uniformizing maps converge uniformly up to the boundary (radial
identification of source annuli when needed), every cap mark converges to
the prime end at its slit tip, and if m_q, m are the continuum critical
Ising one-point functions then for every K compactly in D,
sup_{z in K} |m_q(z) - m(z)| -> 0. (9.4)
Proof. Prove along an arbitrary null sequence q_n -> 0.
Connectivity of filled complements. In the straightened chart the filled
complement is a base region with finitely many thin rectangular teeth;
nearby points in one tooth or in the base join by a short path of diameter
at most a fixed multiple of their distance; near a junction the finitely
many fixed templates have a finite local quasiconvexity constant (scale
invariant); nonincident pieces have fixed positive separation. So there are
C, t_0 > 0 independent of q such that points at distance t < t_0 in one model
filled complement join there by an arc of diameter <= C t. With H the fixed
chart homeomorphism and J_q the polygonalization/smoothing, the physical
complements are images under H_q = J_q o H, and H_q, H_q^{-1} tend uniformly
to the identity; so they share moduli of continuity mu, nu on relevant
compacts and physical points at distance t in one complement join by an arc
of diameter at most theta(t) = mu(C nu(t)) -> 0. (The fixed other complement
in the annular case has the same by its Jordan parametrization.) This is
required for the FILLED complements, not the boundary curves.
Interior uniformization. [Papamichael 2008, Theorems 1.2.1 and 1.3.1] give
the disk and annulus uniformizations and uniqueness of the annular radius
ratio. Labelled complements converge in Hausdorff distance and compacts of
D lie eventually in D_q. The normalized kernel theorem [Carmona-Fedorovskiy
2024, Def. 2.8 and Thm. 2.9] gives interior convergence for disks. For
annuli write the source as A(r_q, 1), preserving labels ([Papamichael 2008,
Thm. 1.3.1]); a fixed essential annulus compactly in D lies eventually in
every D_q and every D_q lies in a fixed bounded annulus with inner disk in
the inner complement (both inclusions essential; the intervening annuli
separate the same two complements), so monotonicity of annular modulus
[Beliaev 2015, Prop. 4.3.5 and Sec. 4.3.4] gives 0 < r_- <= r_q <= r_+ < 1.
For a subsequential limit r_q -> r in (0,1), let g_q : D_q -> A(r_q,1) be
inverse uniformizers, F_q = g_q^{-1}, normalized (inverse at fixed z_* in D).
Take simultaneous locally uniform normal limits g on D and F on A(r,1). The
inverse maps have winding one on a fixed essential curve and are bounded
away from zero, so the limit F is nonconstant (a nonzero constant has
winding zero, and r < |g| < 1 by max/min principle). Local convergence in
F_q(g_q(z)) = z gives F(g(z)) = z. To exclude values on the limiting slit
boundary: for a in C minus D, Hausdorff convergence gives a_q in C minus D_q
with a_q -> a; F_q - a_q is zero-free; Hurwitz plus nonconstancy of F gives
F(w) != a for all w in A(r,1). So F(A(r,1)) is in D; applying local
convergence to g_q(F_q(w)) = w gives g(F(w)) = w. Thus F, g are conformal
inverses; annular modulus is unique (identifies r) and normalization
identifies the maps; every subsequence has the same limits, proving interior
convergence.
Boundary equicontinuity. Let F_q be a forward map from the unit disk or the
source annulus. [Papamichael 2008, Thm. 1.2.1-1.2.2] give disk
uniformization and the Jordan boundary extension; [Thm. 1.3.2] gives the
annular extension preserving labels. So each map extends continuously to its
boundary, its image is bounded, and univalence gives int |F_q'|^2 dA <= A_0.
Fix a source boundary point zeta and R smaller than the source collar width
and dist(normalizing point, boundary). For 0 < r < R the arcs
C_t = {|z - zeta| = t} (r < t < R) are crosscuts with endpoints on one
boundary component. With ell_q(t) = length F_q(C_t), Cauchy-Schwarz and
polar integration give int_r^R ell_q(t)^2 / t dt <= 2 pi A_0; so some
t in (r,R) has ell_q(t) <= s(r) = (2 pi A_0 / log(R/r))^{1/2}. Join the
image endpoints inside their common filled complement by a simple arc of
diameter <= theta(s(r)); it meets the image crosscut only at its endpoints,
and their union is a Jordan curve in a ball of radius s(r) + theta(s(r))
about one endpoint (small r excludes z_*, and in the annular case also the
separation of the complements). The source cap cut off by C_t misses the
source normalizing point, so its image lies on the small bounded side (the
other side contains z_* and the curve meets the domain only in the
crosscut). Hence diam F_q(B(zeta,r) cap source domain) <= 2[s(r) +
theta(s(r))] -> 0 uniformly in q and zeta. With interior convergence this
gives uniform convergence on closures. (Annuli: use radial maps between
A(r,1) and A(r_q,1); r_q -> r in (0,1) makes these uniformly Lipschitz and
uniformly close to the identity.)
Marks. A finite simple slit tip has exactly one prime end above it: in the
chart a small disk about the tip is cut along a segment ending there, and
nested crosscuts around the endpoint give its unique approach class. A root
has two prime-boundary preimages but no sign change sits at a root. If b_q
is a cap mark, any limit of F_q^{-1}(b_q) maps to the tip by uniform
convergence, and the unique prime-end preimage identifies it. A base mark
away from roots is unique because the base is Jordan and no slit is
incident there. Distinct marks give distinct limiting prime ends, so all
sign arcs stay nondegenerate.
Spin means. [CHI 2021, Theorem 1.3], with convergence per Definitions
3.1-3.2, applies to the fixed smooth/polygonal domains here and to the
limiting marked disk/annulus (boundary arcs read on the prime boundary
through the conformal pullback of Definition 3.6). The mark argument keeps
the number of boundary components, the cyclic arc order, and the
nondegeneracy of every sign arc. The rough-boundary local normalizations of
[CHI 2021, Lemma 3.34] cancel in the correlation ratios in the proof of
Theorem 1.3; bulk convergence is uniform on compacts. Continuity of the
continuum correlations under marked convergence follows by a diagonal
argument: if (9.4) failed, choose an offending sequence of continuum
outlines and then a lattice approximation to each so fine that its
normalized mean is within 1/k of that continuum mean and close in the marked
topology; the combined lattice sequence approximates the limiting marked
domain, contradicting the same bulk convergence theorem. []
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
PART 9 (continued 9B-1). INCIDENCE CRITERION AND OUTER-WALL WEIGHT
--------------------------------------------------------------------------------
9.2b The boundary-incidence criterion. Marked convergence controls the
means for REGULAR comparison outlines. To compare an IRREGULAR realized
wall with them we need a DISCRETE ordering valid for every nearby
transcript. It comes from the signs the common target component meets.
All models below are represented on a common finite nearest-neighbour
graph by declaring some vertices pinned; vertices beyond a complete
boundary layer can be assigned arbitrarily, since that layer separates them
from the target.
Lemma if:incidence-order (ordering by boundary incidences). Let mu_0, mu_1
be two ferromagnetic Ising laws with the same interactions and possibly
different pin sets. Let C be the component of the vertices FREE IN BOTH
models that contains a specified connected target. Suppose that for every
edge xy with x in C and y not in C, at least one of the following holds:
(1) y is pinned minus in model 0;
(2) y is pinned plus in model 1;
(3) y is pinned to the same sign in both models.
Then mu_0 restricted to C is stochastically dominated by mu_1 restricted to
C (mu_0|_C <= mu_1|_C). The same holds for a union of target components with
the analogous incidence condition.
Proof. Condition separately on all spins outside C. A boundary neighbour y
cannot be free in both models (its edge to C would put it in that common
free component), and each alternative gives sigma_0(y) <= sigma_1(y) for
every pair of exterior assignments consistent with the pins. So at each
x in C the induced boundary FIELD in model 0 is at most that in model 1,
with identical internal ferromagnetic couplings on C. Monotone coupling of
the finite heat-bath chains on C (equivalently, ferromagnetic boundary
monotonicity) gives domination of the marginals on C for each pair of
exterior assignments; mixing them proves the claim. []
Lemma if:triangle-incidence (examined triangles preserve the facing sign).
Fix an admissible limiting datum and disjoint small neighbourhoods of its
roots and tips. For sufficiently close resolved data and sufficiently fine
mesh, an edge from a free vertex in the target component to an EXAMINED
neighbour, OUTSIDE those neighbourhoods, sees the prescribed sign of the
corresponding ACCESSIBLE face of the ideal datum. The short approach to the
realized wall has length O(delta) in the physical plane; identifying it with
a limiting wall face uses the datum tolerance and the chart's modulus of
continuity.
Proof. An edge between free vertices cannot cross a traced zero segment:
the traversed triangle containing such a crossing would have inspected both
endpoints. The same applies to the open part of an edge whose last endpoint
is examined. If that endpoint was inspected in a traversed triangle, append
an approach of length O(delta) in that triangle, on the side of the zero
segment carrying its spin sign (affine interpolation gives a constant
vertex sign on each side). If instead it belongs to the initially revealed
side, the edge lies within O(delta) of the starting polygon; approach the
accessible arc of that polygon directly. Every initial sign switch produces
a zero crossing of the starting contour and is explored, so no additional
switch lies on a retained base arc away from its roots.
Outside the node neighbourhoods use the chart of Lemma 9.2. The limiting
edge lies in a strip disjoint from all nonincident graph pieces. Uniform
parametrized convergence places the perturbed strand in a narrow middle
substrip and makes it a crosscut between the two plugged ends. A simple
crosscut separates the long sides even if it repeatedly backtracks, so the
approach just described cannot change sides without crossing this wall; its
sign is the sign of that accessible face. The positive physical separation
(after fixing chart and node neighbourhoods) absorbs the O(delta) detour.
Only near a stopping tip could the short approach slip around an untraced
end -- exactly the excluded neighbourhoods. []
9.3b A common weight for the distant wall.
Lemma fd:outer-wall-weight. Let C and W be nested lattice square contours
with a fixed positive macroscopic collar between them (the critical
nearest-neighbour square lattice). Let T be strictly inside C, and let
nu_1, nu_2 be two laws with plus spins on W. They may have different
prescribed spins inside C but the SAME interactions and pinning rule on and
outside C, where C is free. Write nu_i^b for nu_i after replacing the plus
wall by any assignment b on W. Then there is a positive function w_b of the
spins on C, independent of the data strictly inside C, such that
d nu_i^b |_{T union C} = ( w_b d nu_i |_{T union C} ) / nu_i(w_b),
sup w_b / inf w_b <= K, (9.5)
K depending only on the collar (uniformly in mesh and b). Consequently
|| nu_1^b|_T - nu_2^b|_T ||_TV
<= 2 K || nu_1|_{T union C} - nu_2|_{T union C} ||_TV. (9.6)
Proof. For a contour assignment j, sum all spins strictly between C and W,
keeping j on C and b on W; call this exterior partition weight A_b(j) (use
the SAME allocation of energies along C in every law). The quotient
w_b(j) = A_b(j) / A_+(j) involves no prescribed spin or interaction strictly
inside C, and factoring the Gibbs density at C gives (9.5) (normalized
separately for each nu_i).
Consider the Ising law on the collar alone with both contours free; its
joint marginal on (sigma_C, sigma_W) is proportional to A_b(j). So for any
j, j',
log w_b(j) - log w_b(j')
= log A_b(j) + log A_+(j') - log A_+(j) - log A_b(j')
is a log-odds rectangle, which Lemmas 2.2 and 2.3 bound by
4 arctanh q0 <= a geometric constant (the fixed square collar makes q0
bounded below by a fixed amount, as in Corollary 3.4). Exponentiating gives
the oscillation bound on w_b; energies involving only wall spins multiply a
partition weight by a scalar and do not affect it.
Rescale so 1 <= w_b <= K, and put d = ||nu_1|_{T union C} -
nu_2|_{T union C}||_TV. For an event A of these joint spins, the layer-cake
formula gives |nu_1(w_b 1_A) - nu_2(w_b 1_A)| <= K d and
|nu_1(w_b) - nu_2(w_b)| <= K d, hence
| nu_1(w_b 1_A)/nu_1(w_b) - nu_2(w_b 1_A)/nu_2(w_b) | <= 2 K d.
Sup over A, then restrict to T. []
Application to outward erosion. It suffices first to prove Lemma 9.1
(if:erosion) for a CONSTANT plus distant wall. Choose a square contour C0
between the full compact target T and the distant wall W, with positive
clearances from both and enclosing the whole permitted band. All three inner
laws mu^0, mu^+, mu^f share a free collar from C0 to W. Apply the
constant-wall erosion construction to the enlarged target T union C0 with
error budget smaller than eps/(2K); the extra contour keeps positive
distance from band and W, so the geometric hypotheses hold. For any b on W,
(9.6) transfers each of the two joint errors to T, their sum still < eps,
the weight being the same for every transcript and both insertions; the
fully inserted law stays determined only by the barrier, the mesh, and b.
Adding plus then minus pins keeps the two orders. Constants and barrier are
uniform in b, and no limiting description of a mesh-dependent wall is used:
marked-domain convergence is applied only with a constant wall. []
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
PART 9 (continued 9B-2). PROOF OF LOCAL EROSION; BARRIER BUFFER
--------------------------------------------------------------------------------
9.4 Proof of Lemma 9.1 (if:erosion). By the outer-wall reduction (9.3b) take
the distant wall constant in the outward case. Keep the source and test
contours fixed. We build the barrier and the two mean bounds used in (9.3),
choosing things in the order their uniformity requires: node
neighbourhoods, comparison outlines, erosion contour, then datum tolerance
and mesh.
STAGE 1: node neighbourhoods and their mean error (plugs). Use the chart of
Lemma 9.2. Choose disjoint small closed neighbourhoods P_1,...,P_k of the
roots and tips; include one auxiliary node on the base if needed to cut its
circular strip into finitely many coordinate strips. Each is the image of a
small rectangle in the chart. Choose a smaller rectangle strictly inside
each one, so every incident ideal edge is monotone in its local strip
coordinate until it leaves the larger rectangle (nested rectangles leave
positive slack for the joins below); compact parts of nonincident arcs have
positive separation.
In a PLUGGED law fix every lattice spin in these neighbourhoods to the SAME
assignment in every law under comparison (overwriting any old
prescription); denote it by a superscript P. The plugs close the channels
between displaced boundaries; their assignment need not agree with the old
side signs.
For a single plug and any law lambda used below, Corollary 3.5 and the
moment bound (Lemma 4.3) give
| E_lambda M_J^delta - E_{lambda^P} M_J^delta |
<= omega(diam P), omega(t) -> 0, (9.7)
uniformly in all other pins and for fine meshes. Indeed, the contracted
connection probability in Lemma 2.3 is symmetric in its terminals, and its
small-patch bound controls the density on the remote contour J^delta when
the assignment in P changes: if that density ratio is R_P with
e^{-b_P} <= R_P <= e^{b_P}, b_P -> 0 as diam P -> 0, then
|E[(R_P - 1) M_J^delta]| <= (e^{b_P} - 1)
(E(M_J^delta)^2)^{1/2}. When spins in P were free, condition on their
values, apply the uniform comparison, average; to overwrite old pins,
remove their prescriptions while keeping vertices and edges, same
conditional comparison. Pins outside P are common, and J keeps its fixed
free collar, so has a uniform moment bound. Sum over the finitely many
plugs and choose their sizes so the sum is below a small a > 0 for every
law used (a will be fixed by the transmission constant and the requested
error).
STAGE 2: regular upper and lower boundaries. Build deterministic comparison
laws indexed by a small chart displacement q > 0. On a one-sided base arc
move a plus face toward the fresh side and a minus face away by amounts
comparable to q. On a slit move a thin ribbon entirely toward the plus side
by ~ q, keeping the respective plus/minus face signs; its thickness is a
small fixed fraction of q. These give the UPPER outline; reverse both moves
for the LOWER outline. Join pieces near roots and tips by fixed scaled
polygonal templates, one sign-switch mark per cap, no narrowing neck below a
fixed fraction of the template scale. For small q all joins lie in the
smaller node rectangles. Regularly approximate each outline by Lemma 9.3
within a smaller fraction of its clearances. Let nu_q^u, nu_q^l be the
unmodified Gibbs laws (same distant wall in the annular case).
Lemma 9.5 (if:marked-convergence) makes their continuum one-point functions
converge to the same limit as q -> 0. For fixed q, [CHI 2021, Theorem 1.3]
gives convergence of the normalized lattice means uniformly on J. Since
J^delta has O(delta^{-1}) vertices,
lim_{q->0} limsup_{delta->0}
| E_{nu_q^u} M_J^delta - E_{nu_q^l} M_J^delta | = 0. (9.8)
Choose q and its regular approximations so this gap is small for all
sufficiently fine meshes (after the plug sizes), keeping every join strictly
inside its plug.
STAGE 3: the erosion contour and its free neighbourhood. Now choose
0 < rho << q. At chart distance rho on the fresh side put an outline
parallel to the base and around BOTH sides and the tip of each slit.
Prescribe the face signs, one switch per cap. This is a Jordan contour; a
regular approximation gives Gamma_d with switches on grid-aligned straight
portions. For fine mesh a simple lattice cycle following it, with a boundary
layer if needed, has the prescribed cyclic arc order and separates the
target from the old boundary.
After rho and this approximation are fixed, the new layer has positive
physical separation from the ideal datum; choose the datum neighbourhood and
mesh small enough to preserve it. The recorded lattice datum uses R_delta
(undoing it moves curves by only O(delta)); uniform continuity of the fixed
chart and its inverse turns a small recorded-data tolerance, followed by a
fine mesh, into the required chart tolerance for the actual wall. Absorb
this and the examined triangles' O(delta) thickness in the remaining
clearance; the ENTIRE new layer is then free in the original conditional
law.
INWARD case keeps more. Let D be the base interior with slits removed; the
outline wraps around every slit, so closure(Int Gamma_d) is inside D, and
with a thin exterior collar it is a compact set C contained compactly in D.
Uniform parametrized convergence of the base preserves its winding number
about every point of C (C has positive distance from the base); nearby slits
stay away from C too. So C lies in the accessible domain of every
sufficiently near datum; nonvisible traces and hidden pockets lie in its
blocked complement. All examined vertices are on the initial side or within
O(delta) of the traced boundary, so the whole component kept inside Gamma_d,
with a positive exterior collar, is free of old pins, uniformly over the
datum neighbourhood. (See Figure 9.F.)
Figure 9.F (ordered comparison in the slit chart). The old slit has plus
on its left, minus on its right, placed at chart coordinate x = 0; the
black tested wall is at x = 0. In (a) the UPPER ribbon moves toward the
plus side; in (b) the LOWER ribbon moves toward the minus side. Common
root and tip plugs seal each interwall channel; dotted cap traces show
geometry overwritten by the common plug. The common free target component
sees (a) an upper plus or tested minus face, or (b) a tested plus or
lower minus face, or identical plug data. Dashed erosion layers
(chart coordinate -rho, +rho) may supply the tested frontier after
insertion; the opposite layer is screened. Lower strip: shared source
contour S_0, intermediate cut J, target T; all changed facing pins
precede S_0, and for each fixed distant wall the continuation beyond it
is the same. Coordinates are chart coordinates with q >> rho >> xi; no
metric regularity of the chart is assumed.
STAGE 4: incidence ordering for every nearby transcript. Let mu^0, mu^+,
mu^f be the three laws. Compare their plugged versions to the same two
plugged regular laws. Consider a slit strip outside the node plugs: give
the old slit plus on its left, minus on its right, put it at x = 0. The
actual perturbed strand lies in |x| < xi with xi << rho; the upper ribbon is
near x = -q, the erosion plus layer near x = -rho, the erosion minus layer
near x = +rho.
Parametrized convergence also controls visits to the strip ends. For each
ideal edge its part between the smaller node rectangles lies in a strip
separated from all nonincident edges; on the matched perturbed edge select
its last exit from the first smaller rectangle before its first entry into
the second. This is a simple crosscut. The ideal edge is monotone near each
node, so closeness of parametrizations places these exit/entry parameters
near the ideal ones, and the positive gap between the smaller rectangle and
the actual plug keeps the discarded prefix/suffix inside their plugs (in a
local edge coordinate with exit at s = 2h and plug exit at s = 3h, a
tolerance e < h/3 puts the prefix at s <= 2h + 2e < 3h). Retain the whole
crosscut, including later visits to an outer plug, so every portion of the
original edge outside the actual plugs is retained. Turns and repeated plug
visits do not change the crosscut's separation of its long sides (filling
plugs can remove components but cannot join the two sides). Its orientation
identifies the accessible face signs, as in Lemma if:triangle-incidence. This
applies to every slit and to each base arc between consecutive nodes;
finitely many edges, so one tolerance works for all, and taking the mesh
below the remaining slack prevents a lattice edge from slipping between a
wall end and its plug.
The upper ribbon and this old-wall crosscut, joined through the wholly
pinned root and tip plugs, close the channel between them, which then cannot
meet the target component of the sites free in both laws; additional
examined pieces cannot reopen it. So the common target component sees only
the plus face of the upper ribbon or the minus face of the old wall. In the
fully eroded tested law the latter frontier may instead be the erosion minus
layer; in the plus-only tested law its extra plus pins lie in the screened
channel. The SAME upper comparison works for all three laws. Lemma
if:triangle-incidence identifies each accessible examined-neighbour sign
with the appropriate wall face (recorded-data displacement already absorbed
in the slack); at plug incidences signs agree by construction.
Base arcs are one-sided. If an arc's sign is plus, the upper outline is
nearer the fresh side, so the upper model supplies a plus pin at the common
component's frontier; if minus, the tested wall is nearer and supplies a
minus pin. Sign-changing joins are inside the common plugs. So every
incidence satisfies Lemma if:incidence-order, and
(mu^j)^P <= (nu_q^u)^P on the common target component, j in {0,+,f}.
For the lower comparison move the ribbon to the minus side: accessible
incidences are the lower minus face or the tested plus face; extra tested
plus pins reinforce the order; erosion minus pins lie in the screened
channel; base-arc argument reversed; plugs give identical signs. The
incidence criterion gives the full sandwich
(nu_q^l)^P <= (mu^j)^P <= (nu_q^u)^P, j in {0,+,f}. (9.9)
STAGE 5: mean bounds and the completed law. Removing the plugs changes each
mean by less than a, so (9.9) places every E_{mu^j} M_J^delta between the two
unmodified regular means with error at most 2a at either end; each ordered
mean difference in (9.3) is at most the absolute regular mean gap plus 4a.
Choose a first, then q so that (9.8) plus this error makes the sum of target
total-variation changes less than eps. All later choices (rho,
regularization, neighbourhood, mesh) preserved these bounds, and NO
spin-scaling limit in a realized transcript's irregular domain was used.
Finally the completed layer separates the target from every old examined
vertex; conditional factorization of the Gibbs weights leaves exactly the
law with the chosen new boundary and the unchanged distant wall, independent
of the transcript. The argument conditions on the full transcript including
auxiliary radii when random. Omitted initial spins and nonvisible traces lie
on the blocked side of the new barrier; a stopping radius with no visible tip
adds no boundary piece. So the same completed law works even when the datum
does not record its numerical radii; averaging over these radii or other
omitted information preserves the two total-variation bounds. []
Lemma if:barrier-buffer (moving a completed barrier outward). In the inward
setting let a regular signed contour Gamma already be fully pinned, and let
V^delta be the entire free component it encloses (pinned layer excluded).
Assume a positive exterior collar of Gamma contains no other pins (as
supplied by Lemma 9.1). For any compact target on its inner side and any eps
> 0 one can choose a regular surrounding contour Gamma-hat at positive
distance from Gamma, with corresponding signs, so that on the marginal on
V^delta the two steps (remove the old plus pins on Gamma; then remove the old
minus pins) EACH change the target law by less than eps. The farther pins on
Gamma-hat are present from the start and are initially harmless; step 1 is
downward and step 2 upward; after both, V^delta is separated from every fixed
pin by a positive macroscopic distance, uniform for fine meshes.
Proof. Choose Gamma-hat a sufficiently thin outward offset of Gamma matching
the finite sign arcs, and put the new pins there before changing the old
ones; the full old layer disconnects the kept volume, so the farther pins are
initially harmless. Keep the SAME V^delta throughout (do not first discard a
free strip inside the old layer). Use the construction of Lemma 9.1 with no
slits: small identical plugs around the finitely many sign switches, upper
and lower regular outlines by moving plus/minus arcs in their ordered
directions, compare via Lemma if:incidence-order. The test contour lies
strictly inside Gamma with a common free collar. As the offset -> 0 both
unmodified envelope means have the same limit (Lemma 9.5); plug errors ->
0 (Corollary 3.5). This controls the means for the original law, the law with
only the old minus pins remaining, and the law with all old pins removed;
the two actual removals are ordered, so Corollary tr:ordered-tv proves the
claims. Fix the small offset afterward; its physical distance from Gamma is
positive and mesh rounding keeps a fixed fraction. The kept volume is inside
Gamma, the effective boundary pins are on the farther contour, and the
original pins are farther still. []
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PART 10. FINITE STOPPING-BAND DESCRIPTORS
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Local erosion (PART 9) assigns a barrier to a NEIGHBOURHOOD of each
admissible datum. Now choose FINITELY many of these neighbourhoods covering
all but a prescribed amount of the reference probability. This is a finite
approximation IN PROBABILITY; the whole space of slit data need not be
compact, and no deterministic bound on the number of slits is required.
Theorem if:band-transfer (stopping-band transfer). Fix an inward or outward
band, target, and ambient square chart as in PART 8, with all stated positive
clearances, and use the critical plus-boundary law in the fixed larger
reference square. For every exceptional probability eps > 0 and target error
eta > 0, generic contour levels can be chosen in any prescribed separated
subwindows so that, for all sufficiently fine meshes:
(1) The stopping observation of Lemma 8.5 (if:stopping) has a FINITE
descriptor Z^delta in {0,1,...,m}, computed from the BAND SPINS. State 0
is exceptional and has probability at most eps in the fixed reference
chart.
(2) Every good state i has a DETERMINISTIC regular signed barrier Gamma_i.
For every realizable transcript with Z^delta = i, adding Gamma_i's plus
pins and then its minus pins changes the target law by a sum of at most
eta in total variation. The target-side Gibbs law Q_{i,delta} is the
SAME for every such transcript, uniformly over all bounded observables on
the target.
(3) In the inward case one may also move the full barrier to a slightly
farther signed contour, keeping the entire enclosed free component as
the kept volume; counting the two old-pin removals too, the sum of
target errors is still at most eta. The final kept volume has a positive
macroscopic buffer from its fixed boundary. All geometric clearances
have positive minima over the good states.
(4) For a fixed finite list of bands the descriptors have CONVERGENT joint
laws; the bins can be chosen to be continuity sets of the limiting data
law, so this finite-space convergence is in total variation.
The number of states, their barriers, and their positive clearances depend
on the fixed geometry and on eps, eta, NOT on the mesh; the mesh threshold
is chosen afterward. If another law (possibly conditional on remote pins) has
band density at most K times that of the reference chart by Corollary 3.4,
its exceptional probability is at most K eps; same for mixtures of such
conditional laws.
Proof. Choose generic levels by Lemma 8.8 (if:visible-data). At each
admissible limiting datum apply Lemma 9.1 (if:erosion), and, for the inward
version, Lemma 9.6 (if:barrier-buffer). Allocate the requested total-
variation budget among the finitely many ordered steps. This produces an
OPEN neighbourhood and a barrier construction valid UNIFORMLY for every
realizable datum in that neighbourhood on all sufficiently fine meshes.
Refine this cover to balls whose closures lie in such neighbourhoods and
whose boundaries have limiting probability zero. For a fixed centre only
countably many radii can give a positive boundary mass; separability
(Remark 8.9) supplies a countable such subcover. By continuity of
probability along increasing finite unions, take finitely many balls whose
union has limiting probability larger than 1 - eps/2. Order them and form
DISJOINT bins by assigning a datum to the first ball containing it. Their
boundaries lie in the finite union of the selected sphere boundaries, hence
have probability zero; the remaining bin is exceptional, and convergence of
the data makes its lattice probability at most eps for all sufficiently fine
meshes.
Associate to a good bin the barrier of the neighbourhood containing its
ball. The local lemmas hold for EVERY transcript in that bin, with no
dependence on an observable. Take the minimum of the finitely many positive
clearances and mesh thresholds. This proves (1)-(3). For a finite list, do
the same in each coordinate and use Lemma 8.8 jointly; every descriptor
string is a continuity set, giving convergence of all its finitely many
probabilities. Finally the bad event depends only on band spins, so the
density comparison gives the last assertion directly, including after
conditioning on the stated remote information. []
Corollary if:descriptor-averages (lookup approximation of conditional
averages). In the outward setting, let F^delta be the stopping information.
For any family of target observables h_delta with |h_delta| <= H, define
a_i(delta) = E_{Q_{i,delta}} h_delta for i >= 1, and choose any exceptional-
bin value in [-H, H]. Then on every good transcript
| E[h_delta | F^delta] - a_{Z^delta}(delta) | <= 2 H eta, (10.1)
and in the reference chart
E | E[h_delta | F^delta] - a_{Z^delta}(delta) |^2
<= 4 H^2 (eta^2 + eps). (10.2)
The good bins, barriers, and mesh threshold work for ALL these observables
simultaneously. Neither the observables nor their lookup values need
converge as delta -> 0. More explicitly, fix a reference continuation law nu
and regard f_delta^nu(T) = E_nu[h_delta | F^delta = T] as a function of the
stopping transcript T; evaluating this SAME function under any buffered
band distribution with density bound K gives the integrated estimate with
exceptional term K eps. It is the REFERENCE conditional-average function
that is transferred; it equals the conditional average for the new law when
that law has the same continuation and distant wall given T. If the distant
wall is changed to an assignment b, the uniform wall extension in Lemma 9.1
supplies completed laws Q_{i,delta}^b, and the conditional averages use the
separate lookup values a_i^b(delta) = E_{Q_{i,delta}^b} h_delta.
Proof. The conditional target law of a stopping transcript is its Gibbs
continuation (Lemma 8.5). On a good bin its total variation distance from
the single law Q_{i,delta} is at most eta; the defining inequality for total
variation gives (10.1). On the exceptional bin both quantities are bounded by
H in absolute value; squaring and integrating gives (10.2). All
constructions in Theorem 10.1 preceded the choice of h_delta. The pointwise
good-bin estimate and the transferred exceptional probability prove the last
assertion. This is why mesh-dependent likelihoods can be transferred without
being assigned a continuum limit. []
Corollary if:random-levels (random levels and finite ball probes). Theorem
10.1 and Corollary 10.2 remain valid when the two contour radii are drawn
with a BOUNDED DENSITY in a fixed compact rectangle of separated radii
strictly inside the permitted windows; different band observations may use
INDEPENDENT draws, and their joint data laws converge, including the
auxiliary randomness.
Further, fix a countable dense collection of marked rational-walk prototypes
in the ambient space X^ of Definition 8.7, and intersect probe balls with
X_band. Allow a probe to test whether the data lie within distance r of one
prototype, r drawn UNIFORMLY in a rational interval with closure in (0,1).
For any prescribed eps, eta, finitely many such probes suffice to obtain the
lookup approximation of Corollary 10.2 with adjusted exceptional probability
at most eps. The finite probe list depends only on the reference local
geometry, the radius rectangle, and the requested tolerances; it is
independent of the bounded observables and of the size of an ambient buffered
array.
Proof. For almost every radius pair the deterministic continuity argument of
Lemma 8.8 applies. Couple the radius draws identically on all meshes,
independently of the converging loops. The records take values in the union
over the permitted radii of Definition 8.7; the deterministic rounding rule
is measurable in the draws, and the recording maps R_delta have uniformly
vanishing displacement. So the continuous mapping theorem (or dominated
convergence after this coupling) proves joint convergence of the encoded
data and the radius draws, and the full stopping information includes those
draws. The compact radius rectangle gives uniform positive separation of
contours and uniform clearance from the window edges, so the local erosion
construction still has a neighbourhood for every admissible resulting datum.
For each datum choose a neighbourhood U_d as in Lemma 9.1. By density there
are a prototype p and rationals 0 < a < b < 1 with d in B(p,a) and
B(p,b) inside U_d. Choose a rational threshold interval strictly between a
and b: a positive test certifies membership in U_d, while every datum in
B(p,a) tests positive for every threshold in that interval. The inner balls
form an open cover of the admissible data; select finitely many covering
probability at least 1 - eps/2 in the reference law, and decode by taking the
first positive test in their fixed order (else the exceptional state). For a
positive test use the single barrier law of its neighbourhood U_d, so (10.1)
holds WITHOUT dependence on the random threshold or on h_delta. Every probe
tie has probability zero (its threshold is an independent continuous
variable, given the data), so the finite string of bits converges in law and
the exceptional probability is at most eps for fine meshes. Only finitely
many entries of any fixed dense enumeration were used; for finitely many
radius windows take the union of the lists. The choices were made in one
fixed local reference chart, and density comparison transfers the error
bounds to other buffered chart laws with the same fixed multiplicative
constant, regardless of the number of tiles. []
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PART 11. THE MAIN THEOREMS, ASSEMBLED
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We now restate the two headline results of the paper, with all the
intermediate machinery of PARTS 1-10 in hand, and confirm each hypothesis.
Theorem in:comparison (finite-graph comparison; = Lemma 2.3). Let G be a
finite graph with nonnegative pair couplings and finite real fields. Let
I, J, F be disjoint vertex sets with I, J nonempty, and prescribe common
spins on F. Form a comparison graph by deleting F, contracting I to one
terminal, contracting J to another, and setting all fields to zero. Let q0
be the probability that the two terminals are connected in the random-
cluster model of this comparison graph with cluster weight 2 and edge
parameters p_e = 1 - e^{-2 K_e} (parallel couplings added on contraction).
Then for every configurations b, b' on J and every a on I,
e^{-4 arctanh q0} <= mu(sigma_I = a | sigma_J = b, sigma_F)
/ mu(sigma_I = a | sigma_J = b', sigma_F)
<= e^{4 arctanh q0}. (11.1)
Either conditional law may be replaced by an arbitrary mixture of these
laws. The bound is symmetric in the two terminal sets.
Proof. This is exactly Lemma 2.3, whose full proof is in PART 2. Its inputs
are Lemma 2.2 (extreme rectangles, L <= 4 arctanh q0) and, for the final
inequality, the Ding-Song-Sun covariance inequality (brick C2) and the
Edwards-Sokal identity. []
Theorem in:resolution (finite conditional resolution; = Theorem 10.1 with
the whole apparatus of PARTS 4-9). Fix the band geometry of PART 8 and use
the critical plus-boundary law in a larger reference square. For every
eps, eta > 0 one can choose contour levels in any prescribed separated
subwindows, a finite descriptor Z^delta in {0, 1, ..., m}, and regular
signed barriers Gamma_1, ..., Gamma_m with the following properties for all
sufficiently fine meshes delta.
(a) The exceptional state 0 has probability at most eps.
(b) For every transcript in a good state i in {1, ..., m}, inserting the
plus pins of Gamma_i and then its minus pins changes the target law by
a sum of at most eta in total variation. The final target-side law
Q_{i,delta} is the SAME for every transcript in state i. Consequently
the conclusion holds simultaneously for all bounded target observables,
including mesh-dependent ones.
(c) For any fixed finite list of bands the descriptors have convergent
joint laws.
Proof. (a)-(b) are Theorem 10.1, using the local erosion of PART 9 and the
barrier buffer of PART 9; the finite descriptor of PART 8; the mass and
transmission bounds of PART 5; the moment and covariance inputs of PART 4.
(c) is the finite-list clause of Theorem 10.1, using the joint convergence
of PART 8. []
Why the intermediate results fit together.
- The pointwise comparison of PART 2-3 lets us overwrite the visible slit
boundary with a regular signed barrier. It is what makes the "change the
boundary at one small patch" step cheap.
- The mass and transmission results of PART 5 turn those local overwrites
into a GLOBAL total-variation bound on the target: the changed patches
must move the normalized magnetization M_J of the cut, and the doubled
coupling prices each patch by its counting segment.
- The moment bounds of PART 4 supply a uniform second moment for M_J, so
the transmission bound converts mean control into target control with no
scale-dependent constant. The nearby-line and prediction results (PART 6)
are the quantitative form of the same machinery.
- Part 7 (complete resolution convergence to CLE_3) makes the visible
boundary well-defined and continuous as delta -> 0, so the finite
descriptor of PART 8 is measurable and its law converges.
- Part 8 (stopped interfaces and signed slit data) extracts exactly the
boundary visible from the target.
- Part 9 (regular boundaries and ordered erosion) is the geometric core:
it produces one regular signed barrier valid uniformly over every
transcript near a given slit datum, using the boundary-incidence ordering
to place every nearby realized wall between two comparison outlines.
- Part 10 (finite stopping-band descriptors) is the probabilistic core:
it turns in-probability local neighbourhoods into a finite descriptor
and a single target-side lookup law per bin.
Assumptions and imported bricks. The only unproved inputs are:
(C1) Critical FK-Ising RSW / one-arm bounds [Duminil-Copin, Hongler, Nolin].
(C2) The Ding-Song-Sun Ising correlation inequality with arbitrary fields.
(C3) Chelkak-Hongler-Izyurov bulk one- and two-spin limits.
(C4) Benoist-Hongler convergence of Ising interfaces to nested CLE_3, with
simultaneous common-edge approximation.
(C5) The Edwards-Sokal / Fortuin-Kasteleyn-Ginibre cluster machinery.
(C6) The Karlin-Rinott total-positivity theory on the Boolean cube.
Each is flagged where used.
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