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THE CRITICAL SPIN FIELD OF THE PLANAR XY MODEL
A complete self-contained proof
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Notation and conventions are collected in Section 0. The main
statement is Section 1. Sections 2-9 build the proof; Section 10
assembles the field limit. Every cited external result is quoted
with its exact hypotheses and normalization.
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0. NOTATION
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Let D = (-1,1)^2 be the open unit square. For n >= 1 put
D_n = [-1,1]^2 and n^{-1} Z^2,
D_n^o = D_n and D,
E_n = { {x,y} : x,y in D_n, |x-y| = 1/n }.
An angle theta_x in R/(2 pi Z) sits at each x in D_n. Boundary
angles, i.e. those with x in D_n \ D_n^o, are fixed to 0.
For b > 0 the fixed-boundary (pinned) spin law is
P_n^b(d theta) = (1/Z_n^0) exp{ b sum_{{x,y} in E_n}
cos(theta_x - theta_y) }
prod_{x in D_n^o} d theta_x / (2 pi).
Expectation is E_n^0 (with b = b_c, see Section 2). The center
magnetization is
a_n = E_n^0 cos theta_0 = E_n^0 e^{i theta_0} > 0.
The center point in E_n requires n >= 1.
Lattice Laplacian, unscaled and of unit conductance, acting on
functions vanishing at the boundary of D_n:
(L_n u)(x) = sum_{y in D_n, |x-y| = 1/n} (u(x) - u(y)),
G_n = L_n^{-1} (lattice Green function, no n^2 factor).
Continuum objects. Phi_D is the real zero-Dirichlet Gaussian free
field on D, covariance (-Delta_D)^{-1}. Its kernel is written
G_D(z,w) = (1/(2 pi)) log(1/|z-w|) + R_D(z,w),
where R_D is smooth across the interior diagonal on compact
subsets of D x D. For circle averages of radius eps about z in D,
Phi_{D,eps}(z) = average of Phi_D over the circle |u-z| = eps.
Constants:
b_c = critical inverse temperature (Section 2),
a = a(b_c) = 8 pi (critical Bessel coefficient, cited),
K_* = a/(4 pi^2) = 2/pi (stiffness).
Write H^{-3}_loc(D) for distributions T with chi T (extended by
zero) in H^{-3}(R^2) for every chi in C_c^inf(D), with the usual
local topology.
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1. THE MAIN THEOREM
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The normalized complex spin field is the random distribution
S_n = n^{-2} sum_{x in D_n^o}
a_n^{-1} exp{ (G_n(x,x) - G_n(0,0)) / (2 K_*) }
e^{i theta_x} delta_x,
where delta_x(f) = f(x). The scalar a_n^{-1} is the exact inverse
center magnetization, and the Green factor is a fixed function of
x. Both are the same for every test function and every order.
The target field is the full-variance imaginary chaos
V_* = lim_{eps -> 0} exp{ i Phi_{D,eps} / sqrt(K_*)
+ E Phi_{D,eps}^2 / (2 K_*) },
understood as a complex random distribution tested against smooth
compactly supported functions. The counterterm uses the entire
variance, including its finite spatial part.
THEOREM 1.1. For the fixed-boundary critical XY law with b = b_c,
as n -> infinity the distributions S_n converge in law to V_* in
H^{-3}_loc(D). Moreover, for every k >= 1, every f_1,...,f_k in
C_c^inf(D;C), and all nonnegative integers p_j, q_j,
E_n^0 prod_j S_n(f_j)^{p_j} \overline{S_n(f_j)}^{q_j}
-> E prod_j V_*(f_j)^{p_j} \overline{V_*(f_j)}^{q_j}.
The joint laws of the complex smeared fields also converge.
Nonneutral moments are included: the total number of fields need
not equal the total number of conjugates. The theorem does not
assert an asymptotic formula for a_n, nor does it by itself pass
to the free-boundary infinite-volume correlation.
The proof has nine ingredients, developed in Sections 3-10:
(i) spin-height duality (Section 3);
(ii) finite Gaussian cut-and-restore comparisons (Section 4);
(iii) sector cost at fixed positive-radius holes (Section 5);
(iv) vanishing flat-pin interaction across a long annulus
(Section 6);
(v) gluing kernels and a stability lemma (Section 7);
(vi) separated center-normalized correlations (Section 8);
(vii) a uniform two-point bound up to the diagonal (Section 9);
(viii) the continuum circle chaos and its moments (Section 10);
(ix) collision removal, moment and law identification, local
tightness (Section 10).
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2. THE PHYSICAL CRITICAL POINT
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For R >= 1 let Lambda_R = [-R,R]^2 and Z^2 and let E_R be its
unordered nearest-neighbour pairs. The free-box law at b > 0 is
P_{R,b} = (1/Z_{R,b}) exp{ b sum_{{x,y} in E_R}
cos(theta_x - theta_y) }
prod_{x in Lambda_R} d theta_x / (2 pi),
with no exterior edges and no fixed spins. Define
C_b(r) = lim_{R -> infinity, R >= r} E_{R,b} cos(theta_0 - theta_{r e_1}),
m(b) = lim_{r -> infinity} -(1/r) log C_b(r),
b_c = inf{ b > 0 : m(b) = 0 }.
The limits exist, 0 < b_c < infinity, and m(b_c) = 0; these are
the standard phase-transition facts for the cosine convention
above (van Engelenburg-Lis, Theorem 1, and the normalization
comparison in the companion Bessel paper). No critical exponent
enters the definition of b_c. For b > b_c the mass m(b) > 0; at
b_c it vanishes.
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3. SPIN-HEIGHT DUALITY
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A graph G = (V,E) with an orientation of every edge is fixed for
this section; nabla_e h = h_y - h_x for e = (x,y). An integral
connection is an antisymmetric edge function alpha with values in
2 pi Z. Bessel height weights are
p_b(j) = e^{-b} I_j(b), j in Z,
where I_j is the modified Bessel function. The symmetry
p_b(j) = p_b(-j) makes the choice of orientations immaterial.
A quadratic observation penalty exp{ -|B h - v|^2 / 2 } may be
included, with B a finite real linear map and v a prescribed
real vector. Retained penalties are sums of component-local forms
in every cut topology: each row of B is supported in one
component, where it either annihilates the constant or confines
it.
Convention for constants. A component with no absolute constraint
and no confining observation is summed modulo one common
translation by 2 pi Z. On a component whose constant is confined,
all translation copies are summed. An absolute pin fixes that
constant instead. After an edge cut produces several components,
fixing a representative on each piece does not fix the relative
translations when the pieces are later joined.
Write s for the collection of shifts (the connection alpha and any
observation centers), and let Z_G^s be the partition sum with
counting measure and the preceding convention. Superscript 0
means zero connection and zero centers, all cuts and penalties
retained. The sector ratio is
F_G^s = Z_G^s / Z_G^0.
If data u are prescribed on a set p, Z_G^s(u) is the partition sum
with those data imposed (not their probability). Data may fix
every height on p, or every difference from a chosen site of p
leaving the common height free. Relative-data sets lie in one
component of every topology compared.
Changing alpha by the gradient of a 2 pi Z-valued function and
changing heights by its negative is an exact reindexing (an
integral gauge change); observation centers and prescribed data
must move together. A connection is flat on a region if its
circulation round every contractible cycle vanishes; on a
multiply connected region the remaining circulations are its
periods.
LEMMA 3.1 (duality). Let G_n be the dual square grid: vertices are
centers of lattice squares of [-1,1]^2, edges join neighbouring
centers. Given integers l_x indexed by interior primal vertices,
choose an integral connection s on G_n whose circulation around x
is 2 pi l_x. Then
E_n^0 exp( i sum_x l_x theta_x ) = F_{G_n}^s.
Such a connection exists without a neutrality assumption. If
holes containing the charged vertices are removed, its restriction
to the remaining graph is flat, with periods specified by the
enclosed charges; on that graph the bare partition ratio depends
only on those periods.
Proof. Edges along the spin boundary have both endpoints fixed at
0, so their interaction factors cancel. Expand every other
interaction as e^{b cos u} = sum_{j in Z} I_j(b) e^{i j u}.
Integration imposes divergence conditions only at interior
vertices: prescribed integer charges in the numerator, zero in the
denominator; no condition at a boundary vertex. Hence one may carry
each charge to the boundary by a fixed current even if the total
charge is nonzero. Subtract that current in the numerator and
rotate the remaining edge variables onto dual edges: each retained
primal edge crosses exactly one dual edge, and the elementary dual
cycles are generated by the cycles round interior primal vertices,
so the rotated zero-divergence current is a closed integral form.
Integration along dual paths represents it as nabla h / (2 pi),
unique modulo one common height translation; the subtracted
current gives the connection alpha. The factors e^b replacing
I_j(b) by p_b(j) cancel. Positivity is clear. After holes are
removed, a contractible cycle encloses no remaining charge and a
cycle round a hole has circulation equal to its total charge. Two
integral flat connections with equal periods differ by an integral
gradient, by integrating their difference from a fixed vertex,
so the last assertion follows by gauge change. []
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4. FINITE GAUSSIAN COMPARISONS
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Throughout, graphs are finite with the fixed primary structure.
The affine inequality of the companion Bessel paper already
includes real observation centers and real numerator tilts; we
recall its Gaussian-chain proof, then apply it to cuts, relative
data and released constants.
LEMMA 4.1 (finite comparison rules). Allow nonnegative quadratic
observation penalties and compatible linear equality constraints
on a finite primary height graph. Then:
(i) increasing centered precision decreases centered covariance;
hence in a proper centered law an additional centered
linear equality has larger probability under larger
precision; the same holds in a quotient law for equalities
invariant under the released constants;
(ii) a shifted-to-centered partition ratio is at most one and
decreases when the same nonnegative precision is added in
affine and centered coordinates; shifts may include
integral connections, nonzero prescribed data and real
observation centers; monotonicity persists with a fixed
real linear statistic in the numerator exponent (the bound
by one is not asserted for that tilted ratio);
(iii) equality constraints are limits of increasing quadratic
penalties; free constants can be released when all data and
real tests are invariant under them.
Proof. Let Lambda be a lattice in a real Euclidean space, z +
Lambda a translate, T positive definite. For Q >= 0, t >= 0, and
a fixed real linear functional ell set
R(t) = [ sum_{x in z+Lambda} exp( -<x,(T+tQ)x>/2 + ell(x) ) ]
/ [ sum_{x in Lambda} exp( -<x,(T+tQ)x>/2 ) ].
The lattice-Gaussian inequalities of Regev and Stephens-Davidowitz
(Theorem 2.1 and Corollary 3.2) give that the numerator law has
covariance dominating the centered denominator law (companion
Bessel paper, Lemma 2.1). With m_{z,ell} and Cov_{z,ell} the
numerator mean and covariance and Cov_0 the denominator
covariance,
d/dt log R(t) = -(1/2) Tr Q ( Cov_{z,ell} - Cov_0
+ m_{z,ell} m_{z,ell}^T ) <= 0.
On a proper affine subspace, first complete the square in its
parallel span; the remaining offset is orthogonal for the current
precision T + tQ, so the parallel-span covariance still dominates
while the raw mean picks up the fixed offset. The centered maximum
of a Gaussian lattice sum gives the untilted bound R(t) <= 1. To
obtain the Bessel law, replace each primary edge by N independent
integer increments with normalized weight q_N^{j^2}, q_N = b/(2N)
and physical increments 2 pi j. Uniformly on compact real sets,
( sum_j q_N^{j^2} e^{u j} ) / ( sum_j q_N^{j^2} )
= 1 + (b/N)(cosh u - 1) + O_u(N^{-2}),
whose limit has moment generating function exp{ b(cosh u - 1) },
i.e. tends to the Bessel weight p_b. Matching endpoints, cycles,
periods and prescribed data gives parallel affine lattices for
numerator and denominator; appending the residual coordinates
B h - v realizes arbitrary real centers. Adding a quadratic
penalty or a matching constraint is thus precisely the precision
increase just considered. All limits are on a fixed primary graph:
a spanning forest expresses every height difference as a sum of
full-chain increments, which have uniformly bounded exponential
moments at each fixed real argument, while the other normalized
edge weights are at most one; the confined component constant is
summed against its Gaussian penalty. Dominated convergence passes
the comparisons to p_b (companion Bessel paper, Lemma 2.2).
Equality limits use monotone convergence, only on nonempty affine
fibers. Centered pin monotonicity first uses a finite pin penalty:
the derivative of the log expectation under additional centered
precision Q is (1/2) Tr Q (Cov_unpinned - Cov_pinned) >= 0, then
the pin penalty is taken to infinity. []
LEMMA 4.2 (cut comparison with prescribed data). Let G^cut be
obtained from G by deleting edges, with the same observation
penalties and compatible prescribed data on both graphs. Each
retained observation and each relative-data set is component-local
in G^cut. Whenever the free constants are left free by those data,
Z_G^s(u) / Z_G^0(0_p) <= Z_{G^cut}^s(u) / Z_{G^cut}^0(0_p).
The conclusion also holds for absolute data fixing some constants
and observations confining others, and for a fixed real numerator
tilt on affine edge increments if each deleted increment is
retained independently with its tilt on the cut side. Every real
test on a released component must annihilate its constant.
Proof. For a deleted edge e = (x,y), retain an independent chain
with total increment J_e in 2 pi Z; its normalized sum without
matching is one. Restoring the edge imposes J_e - h_y + h_x -
alpha_{xy} = 0 (the centered version has alpha_{xy} = 0), which
on the enlarged affine coordinates is increasing the coefficient of
the squared residual to infinity. Prescribed-data fibers of
numerator and denominator are parallel lattices; this remains true
for relative data h_z - h_{z_0} = u_z, which are linear equality
constraints. Lemma 4.1 gives the comparison. Integrating the
matched chain contributes p_b((h_y - h_x + alpha_{xy})/(2 pi)) and
all independent-chain normalizers cancel. For a real edge tilt,
use J_e on the enlarged graph: on matching it equals the affine
edge increment of the restored graph, so the tilted comparison
applies with no extra scalar.
Constants. Before comparing, add a site penalty
(delta/2) sum_x h_x^2 and sum all translations. A free component
with m vertices, orbit representative h and mean hbar, contributes
exp( -(delta/2) sum_x (h_x - hbar)^2 )
sum_{k in Z} exp( -(m delta/2)(hbar + 2 pi k)^2 ).
Divide its translation sum by c_{m,delta} = (1/(2 pi))
sqrt(2 pi / (m delta)). Poisson summation shows the quotient tends
uniformly in hbar to one and is uniformly bounded for small delta;
the first factor is at most one and tends pointwise to one. Forest
exponential-moment bounds give domination, also for fixed real
tilts invariant under the released constant. So the regularized
sum divided by one c_{m,delta} per free component tends to the
quotient sum, and within each shifted-to-centered ratio these
component factors cancel (separately on cut and restored sides,
although those graphs can have different numbers of components).
On a component already fixed or confined, the penalty is removed
by dominated convergence with no divergent factor; this also
covers comparing an unconfined ratio with one in which new
observations confine every component, cancellation being done
separately inside each ratio. Semidefinite intermediate forms are
harmless because the auxiliary penalty makes every sum proper
before the limit. Finally take the chain limit at fixed primary
graph by Lemma 4.1. []
Consequently the bare ratios satisfy
0 < F_G^s <= 1, F_G^s <= F_{G^cut}^s, (4.1)
where the connection on the right is the restriction to the cut
graph. Discarding an additional region (first cut its incident
edges, then drop its at-most-one ratio) gives the same domain
comparison.
COROLLARY 4.3 (restoration at prescribed data). For an edge
restoration with absolute or relative data u on p, define
T^s(u) = Z_restored^s(u) / Z_cut^s(u), d = T^0(0_p).
Using quotient constants on free components and summing all
relative integer translations when components join,
0 < T^s(u) <= d.
Proof. The cut comparison reads Z_restored^s(u)/Z_restored^0(0_p)
<= Z_cut^s(u)/Z_cut^0(0_p); rearranging gives the claim. Positivity
and finiteness hold on a finite quotient graph (forest bound, all
Bessel weights positive). A joined component has one global
quotient, so the numerator sums the relative constants the cut
denominator quotients separately. []
The comparison is pointwise in the data. It bounds a restoration
factor by its centered zero-data value; lower bounds come instead
from averaging separated pin comparisons.
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5. SECTORS AT FIXED POSITIVE-RADIUS HOLES
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Let U be a bounded connected polygonal domain with a simple outer
boundary and finitely many simple polygonal holes, all sides
parallel to the axes with dyadic endpoints. Hole closures are
mutually disjoint and strictly inside the outer boundary; every
side has positive length, so U is Lipschitz with no point contact.
The graph G_n(U) is the nearest-neighbour discretization at mesh
1/n with free boundary and coherent array rounding (Section 0).
Fix integral periods round the holes in units of 2 pi, and a
smooth closed one-form dv_0 with these periods on a neighbourhood
of closure(U) (v_0 is a multivalued primitive: local branches
differ by constants, its differential is single-valued; a sum of
integer multiples of angle functions with centers in the holes
provides one). Let alpha_n be an integral connection on G_n(U),
flat off the holes, with the prescribed periods; its sector ratio
is F_{U,n}^s. Changing alpha_n by an integral gradient leaves the
ratio unchanged.
THEOREM 5.1 (sector cost at fixed geometry). At b = b_c and
a = 8 pi,
lim_{n -> infinity} F_{U,n}^s
= exp( -M(U) / (2 a) ),
M(U) = inf_{g in H^1(U)} integral_U | dv_0 + dg |^2 ,
along every admissible mesh-rounding sequence. The energy depends
only on the periods.
The upper bound extends the divergence-free real-tilt argument of
the companion Bessel paper (proof of Proposition 2.6) to this
fixed multiply connected domain. The lower bound adds distributed
confinement, shows the separated zero-pin interactions vanish,
slits every hole, applies the single-valued height limit on the
pieces, and restores the cuts.
PROOF OF THE UPPER BOUND. Choose a smooth stream function psi
constant near each boundary component (possibly different
constants), and put w = nabla-perp psi, so w vanishes near the
boundary. Sample psi on faces of the height graph and take oriented
differences across edges: this gives an exactly divergence-free
edge flow w_n, supported off the boundary, with coefficients
n^{-1} w to first order. The forms dv_0 and alpha_n, integrated
along lattice edges, share cycle periods, so their difference is a
real gradient. Hence
D_n' = sum_e w_{n,e}( nabla_e h + alpha_{n,e} )
= sum_e w_{n,e} alpha_{n,e} -> integral_U w . dv_0,
a deterministic number. Insert e^{D_n'} into the sector numerator.
Partition U into dyadic squares of side d aligned with the
boundary and cut all edges between distinct squares. In applying
Lemma 4.2, retain an independent Bessel increment on each cut edge
and place that edge's tilt there; restoring identifies the
increment with the affine height difference, so the statistic is
unchanged. The normalized tilted sector sum is then bounded by the
product of centered Laplace transforms on the squares. On each
square the connection is an integral gradient and disappears by
reindexing heights; since the tilt is on affine differences, no
scalar tilt factor appears.
Limiting bound on the product. On internal edges replace w_n by
n^{-1} times one constant vector sampled per square, zero on cut
edges. For fixed d the squared error on cut edges is O_d(n^{-1})
(there are O_d(n) such edges, each coefficient O(n^{-1})); the
remaining squared error has limsup -> 0 as d -> 0 by smoothness of
w. The free-square Laplace limit (Section 4) applies to
constant-vector tests, giving total limiting log transform
(a/2) times the Riemann sum for integral_U |w|^2. To keep errors
inside the transforms, apply Holder with exponent p > 1 to the
constant part and bound the transform of the error at the
conjugate exponent by freeing all increments: for independent
increments J_e of law p_{b_c} and uniformly bounded r_e,
log E exp( sum_e r_e J_e ) <= C sum_e r_e^2,
since E e^{t J_e} = exp{ b_c(cosh(2 pi t) - 1) }. Let n -> inf,
then d -> 0, then p -> 1:
limsup_{n} log F_{U,n}^s
<= - integral_U w . dv_0 + (a/2) integral_U |w|^2. (5.1)
Density. The fields w above are dense in the orthogonal
complement of gradients in L^2(U;R^2). Extend a field in that
complement by zero outside U; its distributional divergence
vanishes, so in a larger simply connected box it has an H^1
stream function, constant on every complementary component of U.
Subtract a smooth function with the same constants near these
components; the remainder has zero trace on dU, and H^1_0(U)
approximation proves the density claim. The gradient space is
closed by Poincare modulo constants. If P is the orthogonal
projection onto its complement, M(U) = |P dv_0|_2^2. Optimizing
(5.1) over the dense class, with optimizer a^{-1} P dv_0, gives
the upper bound. []
LOCALIZATION UNDER CELL OBSERVATIONS. Fix one dyadic polygonal
geometry (possibly with finitely many components or cuts),
partition it into dyadic squares Q of side d aligned with all
boundaries and cuts, and write
u_Q = (1/|Q|) integral_Q u,
T_d(u) = a^{-1} integral |nabla u|^2 + sum_Q u_Q^2.
The lattice observation on Q is the average over its rounded array
of vertices, with penalty (1/2) u_Q^2; these control every
component constant. For two positively separated unions of
positive-area dyadic boxes H, p, let D_{n,d}(H,p) be the centered
absolute-zero pin interaction (Section 4) for this law.
LEMMA 5.2 (localization). For every fixed dyadic mixed geometry
and fixed dyadic pin regions H, p,
lim_{d -> 0} limsup_{n -> infinity} D_{n,d}(H,p) = 0. (5.2)
Proof. At fixed d, the pin limit identifies D_{n,d} with the
Gaussian determinant of (4.1): with orthogonality for T_d,
V_Q = { u : u = 0 on Q }^{perp}, Pi_H = projection onto V_H,
C_d = Pi_H |_{V_p}, D_{n,d} -> -(1/2) log det(1 - C_d^* C_d).
We show this determinant tends to one. Equip component H^1 spaces
with norm |u|_d^2 = T_d(u). Rescale by d^{-1}: observation cells
become unit squares, gradient energy is unchanged in dimension
two, and cell Poincare gives, with constants independent of d,
c |u|_{H^1}^2 <= T_d(u) <= C |u|_{H^1}^2. (5.3)
There are N_d = O(d^{-2}) cells. If f in V_p then T_d(f, phi) = 0
for every phi vanishing on p, so f is form-harmonic off p.
Cutoff estimate, including observation terms. Take a Lipschitz
cutoff chi vanishing near p, transition width and Lipschitz
constant bounded in cell units, and test with chi^2 f:
integral nabla f . nabla(chi^2 f)
>= (1/2) integral chi^2 |nabla f|^2
- C integral_{nabla chi != 0} f^2.
On cells where chi = 1 the observation contribution is f_Q^2; on
transition cells it is bounded by C |f|_{L^2(Q)}^2 by
Cauchy-Schwarz. Hence the plateau energy is bounded by the L^2
mass in a fixed enlargement of the transition layer. The test is
admissible at free sides (no boundary value prescribed).
Iterate. Let E_r be the sum of cell energies at cell-centre
distance at least r from p. Choose chi zero up to distance r+2 and
one beyond r+6; by (5.3), enlarging the layer by a fixed number of
cells,
E_{r+10} <= C (E_r - E_{r+10}),
whose iteration gives exponential decay in cell distance. The
fixed positive separation of H and p is of order d^{-1} cells;
multiplying f by a cutoff equal to one near H and supported away
from p gives an extension of its H-data of energy at most
C e^{-c/d} |f|_d^2. Orthogonal projection is the least-energy
extension, so
|C_d| <= delta_d, delta_d = C e^{-c/d}. (5.4)
An operator-norm bound does not control the determinant; a
summable singular-value bound is needed. For an integer m >= 1,
restrict f in V_p by requiring its Neumann cosine coefficients on
every unit cell, for 0 <= k_1, k_2 <= m, to vanish. The
codimension is at most C N_d m^2, and the cell spectral inequality
gives |f|_{L^2(Q)} <= C m^{-1} |nabla f|_{L^2(Q)}. Choose nested
cell-unions N_1 in N_2 in N_3, each containing H, separated from p,
with fixed-width buffers between successive N_j and between H and
the relative boundary of N_1 (neighbourhoods may meet free walls).
Put E_j(f) = sum_{Q in N_j} ( a^{-1} integral_Q |nabla f|^2 + f_Q^2 ).
Use cutoffs equal to one on N_j, supported in N_{j+1}, with
uniformly bounded gradients. The cutoff estimate and the spectral
inequality give, with f the original function in both tests,
E_1(f) <= C m^{-2} E_2(f) <= C m^{-4} E_3(f) <= C m^{-4} |f|_d^2.
Take eta = 1 near H, supported in N_1, with bounded gradient; cell
coercivity bounds the cutoff extension by |eta f|_d^2 <= C E_1(f),
and minimality of Pi_H f gives |Pi_H f|_d <= C m^{-2} |f|_d. The
positive physical separation leaves room for the buffers when d is
small. The min-max characterization of singular values then gives
s_j(C_d) <= C N_d / j (j >= C N_d).
Together with (5.4), splitting the sum at an index of order
N_d / delta_d yields
|C_d|_{HS}^2 <= C N_d delta_d -> 0.
For small d, |C_d| < 1/2 and hence -(1/2) log det(1 - C_d^* C_d)
<= |C_d|_{HS}^2 -> 0. Components contribute independently, so the
proof covers every fixed cut geometry. []
THE CONFINED SECTOR AND SINGLE-VALUED LIFTS. Return to U of
Theorem 5.1. Fix v = v_0 + g with g smooth near closure(U). Choose
real vertex values v_n, in cut coordinates compatible with alpha_n,
so that
v_n(y) - v_n(x) + alpha_n(x,y) = integral_{[x,y]} dv.
These exist because the difference of the two edge forms has zero
cycle periods. Choose them by fixing branch cuts and local branches
of v first, then replacing alpha_n by the gauge-equivalent
connection with matching integral jumps across the cuts; this
gives samples of fixed branches rather than mesh-dependent
additive constants. On any piece where alpha_n is gauged to zero,
the simultaneous change of v_n gives samples of a smooth
single-valued lift of v. Add cell penalties,
(1/2) sum_Q ( h_Q - (v_n)_Q )^2 in the sector numerator,
(1/2) sum_Q h_Q^2 in the centered denominator,
summing all translation copies; denote the resulting ratio by
R_{n,d}^s(v). The affine precision comparison (Lemma 4.1) gives
0 < R_{n,d}^s(v) <= F_{U,n}^s, (5.5)
since the shifted residuals are additional affine coordinates of
the same positive precision (their centers need not lie on the
height lattice).
LEMMA 5.3 (single-valued lift). Let A be a component of a fixed
dyadic mixed geometry, with free boundary, admitting a
single-valued smooth lift v_A of v. Then the mesh limit exists at
each fixed d, and
lim_{d -> 0} lim_{n} R_{A,n,d}^s(v)
= exp( -(1/(2 a)) integral_A |dv|^2 ).
The same holds for finitely many such pieces, with free constants
treated independently.
Proof. Gauge away the connection on A. The ratio becomes a
centered confined Laplace transform in finitely many cell
averages, times the scalar from expanding the shifted penalties.
The mixed height limit gives its Gaussian limit with the constant
integrated; sampled centers converge to cell averages of v_A, and
convergence of transforms holds at these arguments by Holder
equicontinuity from nearby fixed arguments. Gaussian completion of
the square identifies the limit as exp(-I_d(v_A)/2) with
I_d(v_A) = inf_{u in H^1(A)} { a^{-1} integral_A |nabla u|^2
+ sum_Q (u_Q - (v_A)_Q)^2 }.
The determinant is the same in both Gaussian integrals and
cancels. Trying u = v_A bounds I_d(v_A) by its gradient energy. For
a sequence with bounded functional value, cell Poincare gives
|u - v_A|_{L^2(A)}^2
<= C d^2 ( |nabla u|_2^2 + |nabla v_A|_2^2
+ sum_Q (u_Q - (v_A)_Q)^2 ) -> 0.
Weak H^1 compactness and lower semicontinuity give
I_d(v_A) -> a^{-1} integral_A |nabla v_A|^2. Independent components
give the final assertion. []
The remaining obstruction is that U need not admit a single-valued
lift. Remove it by cuts, using Lemma 5.2 to show the cuts change
the confined ratio by vanishing amount.
PROPOSITION 5.4 (confined sector limit). For the observations and
smooth multivalued v above,
lim_{d -> 0} limsup_n | R_{n,d}^s(v)
- exp( -(1/(2 a)) integral_U |dv|^2 ) | = 0. (5.6)
Observation partitions may be aligned with any fixed finite
collection of the cuts, patches and pin bands constructed below.
Proof. Throughout, Z_G^s and F_G^s = Z_G^s/Z_G^0 include the cell
penalties at fixed (n,d,v) on whichever cut topology G is named; in
particular F_U^s = R_{n,d}^s(v). Parameters are suppressed until the
ordered limits are taken.
The cuts and their guards. Connect each hole to a distinct unused
part of the outer boundary by disjoint simple arcs (opening a
channel from one hole preserves connectivity and merges two
boundary components). Choose positive clearances from nonincident
boundaries and previously chosen arcs; approximate the arcs within
those clearances by simple rectilinear paths with dyadic vertices,
retain short normal segments at boundary sides, remove loops.
Endpoints lie in interiors of boundary sides. Narrow tubular strips
with square corners and dyadic widths, mutually disjoint together
with wider disjoint strip neighbourhoods, can then be chosen.
Delete the edges crossing the two sides of each narrow strip. This
isolates the strips and leaves a domain with every hole slit, so
every piece admits a single-valued lift of v. Strips, wider
neighbourhoods and all later bands have fixed positive widths;
choose dyadic observation cells aligned with all their boundaries,
so coherent rounding makes cells lie entirely within pieces.
One strip deletion. Consider one of the finitely many strip
deletions with earlier deletions retained, and let S be the narrow
strip. Call the configurations before and after deletion "open"
and "cut". Inside a wider strip patch W place a guard p: bands on
both sides of S separating every edge to be restored from the
artificial boundary of W, with positive buffers to both, reaching
the original free walls at the two ends. Parallel collars and
square corners give the same at turns. W has a common single-valued
lift before and after the deletion.
For prescribed absolute primary data u on p let
K^s(u) = Z_open^s(u) / Z_cut^s(u), d_0 = K^0(0).
Conditioning on the full guard separates the restored edges from
everything beyond W (no observation cell crosses a guard division),
so K^s(u) is unchanged if the edges across dW are cut and K^s is
computed from W alone. The prescribed-data comparison gives
Z_open^s(u)/Z_open^0(0) <= Z_cut^s(u)/Z_cut^0(0), hence
0 < K^s(u) <= d_0.
Loss inside the patch. Choose a union of positive-area boxes H
containing all endpoints of restored edges and positively separated
from p in the cut patch. Under its centered confined law,
1 >= E_{cut W,0} ( K^0(u)/d_0 ) >= e^{-D_{n,d}(H,p)}.
To verify the lower bound: compare restored partition sums with
guard data u and 0, then add the pin h_H = 0; precision comparison
decreases their ratio. With H pinned, restoration is a fixed
scalar because all restored-edge endpoints are fixed; dividing by
the corresponding cut-patch data ratio gives
K^0(u)/d_0
>= P_{cut W,0}( h_H = 0 | h_p = u )
/ P_{cut W,0}( h_H = 0 | h_p = 0 ).
Averaging and Bayes' rule give the displayed estimate. Lemma 5.2
makes the lower bound tend to one in the order n -> inf, then
d -> 0.
Both versions of W have lifts and the cuts remove only zero-area
interfaces, so Lemma 5.3 gives the same limit for their
shifted-to-centered ratios. Hence
sum_u ( Z_cut W^s(u)/Z_cut W^0 )
( 1 - K^s(u)/d_0 ) -> 0 (5.7)
in the iterated-limsup sense (n -> inf, then d -> 0). Indeed the
sum equals F_cut W^s - F_open W^s E_{cut W,0}(K^0/d_0), an exact
identity, so no approximation of conditional laws at individual
microscopic values u is needed.
From patch to whole domain. Let G be the current topology with the
narrow strip cut and G' the further cut across dW. At prescribed
guard data, precision comparison gives
Z_G^s(u)/Z_G^0(0_p) <= Z_G'^s(u)/Z_G'^0(0_p),
so, with 0_p prescribed zero guard data and denominators summing
over those data,
Z_G^s(u)/Z_G^0
<= [ P_{G,0}(h_p=0) / P_{G',0}(h_p=0) ]
* Z_G'^s(u)/Z_G'^0. (5.8)
The normalization change is controlled: with R(u) the centered
restoration factor from G' to G at guard data u,
P_{G,0}(h_p=0)/P_{G',0}(h_p=0) = R(0) / E_{G',0} R.
Choose a union of boxes H' containing the endpoints of all edges
across the artificial boundary of W, separated from p. The centered
restoration comparison with H' gives E_{G',0} R / R(0) >=
e^{-D_{n,d}^{G'}(H',p)}, and Lemma 5.2 makes this tend to zero
(cells stay within components after the cut and control constants,
so its hypotheses hold for G'). Multiply (5.8) by the nonnegative
1 - K^s(u)/d_0 and sum: on G' the part outside W factors off as a
shifted-to-centered confined ratio, at most one by Lemma 4.1, so
the global loss is at most e^{D_{n,d}^{G'}(H',p)} times the patch
loss (5.7); it tends to zero, as does the centered loss.
Finish. On the current cut graph define
L_s = sum_u ( Z_cut^s(u)/Z_cut^0 )( 1 - K^s(u)/d_0 ),
L_0 = E_{cut,0}( 1 - K^0/d_0 ).
Summing restored numerator and denominator gives the exact identity
F_open^s = ( F_cut^s - L_s ) / ( 1 - L_0 ). (5.9)
Since L_s, L_0 -> 0 in the prescribed order and F_cut^s <= 1, a
strip deletion changes the confined ratio by a vanishing amount.
Finitely many strips. After all deletions Lemma 5.3 identifies the
product ratio with the right side of (5.6), and the energies add
over pieces to integral_U |dv|^2. Restoring the strips proves the
proposition; the raw d_0 cancel in (5.9), so no bound on their size
is needed. []
COMPLETION OF THEOREM 5.1. For every smooth g near closure(U),
combine (5.5) with Proposition 5.4:
liminf_n F_{U,n}^s
>= exp( -(1/(2 a)) integral_U |dv_0 + dg|^2 ).
Restrictions of smooth functions from a neighbourhood are dense in
H^1(U) because U is Lipschitz; let their energies approach M(U).
This is the lower bound in Theorem 5.1, and the earlier upper bound
gives the limit. Every height or pin theorem was used at fixed
geometry and fixed d; all further limits came afterwards. []
============================================================
6. FLAT PINS ACROSS A LONG ANNULUS
============================================================
Fix small eps > 0 and a closed rectilinear polygon P,
homeomorphic to a closed disk, with dyadic vertices and
B(0, 1 - eps) subset P subset B(0, 1 + eps)
(a sufficiently fine staircase approximation to the circle,
symmetric in the four quadrants). All side lengths and passages are
nondegenerate. Write Q_r = (-r,r)^2, choose lambda a sufficiently
large power of 4, and put
A_lambda = lambda P \ (P)^o,
p = closure(Q_{2 sqrt lambda}) \ Q_{sqrt lambda},
H_in = A_lambda and closure(Q_4),
H_out = A_lambda \ Q_{lambda/4}.
Sobolev spaces on A_lambda refer to its interior. The guard p is a
full square band separating the two ends; each end region contains
all vertices next to its annular wall after fine discretization.
Guard and either end are positively separated, finite unions of
positive-area boxes in the mixed geometry, and discretization uses
coherent array divisions so no boundary division identifies pieces
at a point.
Let mu be the centered free height law on this annulus, with one
common translation quotiented out. For nonempty vertex set Q write
[Q] = { h_x = h_y for all x,y in Q }, i.e. all differences in Q
pinned without pinning its common height. For H = H_in or H_out set
Dbar(H,p) = log( mu([H] | [p]) / mu([H]) ).
PROPOSITION 6.1 (separated flat pins). Fix P as above. For either
H, Dbar(H,p) >= 0 and
limsup_{mesh -> 0} Dbar(H,p)
can be made arbitrarily small by taking lambda large. The mesh
limit is taken at each fixed polygonal geometry.
Proof. First estimate the continuum interaction with a confining
box average. Choose a dyadic box in p with side lengths fixed
positive multiples of sqrt(lambda) and let B_0 be its probability
average. Sum all common translations and add the penalty
(gamma/2)(B_0 h)^2, gamma > 0; let D_gamma(H,p) be the interaction
of h_H = 0 and h_p = 0 under this confined law.
Start with gamma = 1. The pin limit identifies the Gaussian
interaction for T(u) = a^{-1} integral |nabla u|^2 + (B_0 u)^2.
Write |u|_T^2 = T(u) and let
V_Q = { u in H^1(A_lambda) : u = 0 on Q }^{perp_T},
C = Pi_H |_{V_p}.
Claim: |C|_{HS} -> 0 as lambda -> inf.
Between p and H lies a full round annulus, a conformal cylinder
[0,T] x (R / (2 pi Z)), T = (1/2) log lambda - O(1),
with the end s = 0 facing p. For the inner end take radii 8 and
sqrt(lambda)/2 with the logarithmic coordinate reversed; for the
outer end radii 4 sqrt(lambda) and lambda/8. Enlarging lambda makes
both choices available with fixed clearances from the bands.
For f in V_p the trace at s = 0 satisfies
| f|_{s=0} |_{H^{1/2}(S^1)} <= C_0 |f|_T, (6.1)
with C_0 independent of large lambda: after scaling lengths by
sqrt(lambda), this circle, the guard and the observation box lie in
a fixed connected annular region; Poincare with the box mean
controls the H^1 norm there including constants, and the trace
theorem gives (6.1). On the side of this circle containing H, f
minimizes the gradient energy with its trace and free walls, with
no observation term.
Let A map a trace at s = 0 to the restriction to H of this energy
minimizer, the target norm being the least T-norm of an extension
of that restriction to the whole annulus, naturally V_H. Estimate A
on a normalized Fourier basis of H^{1/2}(S^1). For the constant
trace the minimizer on the H-side is constant; an extension of its
H-data can be chosen constant beyond s = T, linear across the
cylinder and zero before s = 0, vanishing on the observation box,
with energy O(T^{-1}). Hence the constant mode has image norm
O(T^{-1/2}). For a sine or cosine mode of frequency j >= 1,
normalization in H^{1/2} gives amplitude O((1+j)^{-1/2}). The
Neumann minimum at s = T on the cylinder alone is a lower bound for
the actual minimum; the Dirichlet-zero minimum at s = T, extended by
zero beyond, is an admissible upper trial. Solutions of u'' = j^2 u
give energies with factors tanh(jT) and coth(jT), differing by
O(e^{-2jT}); orthogonality of the minimizer to zero-trace variations
shows it differs from the upper trial by gradient norm O(e^{-jT}).
That difference has zero trace at s = 0, extends by zero toward p,
pays no observation penalty, and (up to sign) is the required
restriction on H. So
|A|_{HS}^2 <= C_1/T + C_1 sum_{j>=1} e^{-2jT} -> 0.
All extensions lie in H^1 (traces agree across interior circles).
Composing with the bounded trace map (6.1) proves the claim for C,
and the determinant in (4.1) tends to one since |C| <= |C|_{HS}->0.
Thus
lim_{lambda} lim_{mesh} D_1(H,p) = 0. (6.2)
Any fixed gamma > 0. Consider the four partition functions defining
the absolute-pin interaction. With p pinned, B_0 h = 0, so changing
gamma has no effect. With neither region pinned, the ratio of
partition functions at precisions gamma and 1 tends to gamma^{-1/2}
by the mixed height limit (fractional mean uniform and independent
of the mean-free field; translation sums of Gaussian penalties are
bounded periodic). With H alone pinned, let x = x(lambda) be the
variance of B_0 under the massless continuum Gaussian law with that
pin. Applying the hard-pin height limit to the bounded weights
exp(-gamma (B_0 h)^2 / 2) and exp(-(B_0 h)^2 / 2) gives ratio
((1+x)/(1+gamma x))^{1/2}; moreover x -> inf, since a function
equal to one on the p-side and zero on the H-side, linear across
the cylinder, has box average one and gradient energy O(T^{-1}), so
x >= c T. Combining the four partition functions yields
D_gamma - D_1 -> (1/2) log( (1+gamma x)/(gamma(1+x)) ), (6.3)
whose right side tends to zero as lambda -> inf. So the absolute-pin
interaction tends to zero for every fixed gamma > 0.
Absolute versus flat pins. In a centered confined law, the log
interaction of two centered quadratic penalties increases when
either precision increases: if the first precision changes by tQ,
differentiating the four log partition functions gives
(1/2) Tr Q (Cov_{first penalty} - Cov_{both penalties}) >= 0 by
centered covariance comparison. Taking equality limits handles
nested centered constraints. Absolute zero pins are flat pins plus
pinning one reference height in each set after the within-set
constraints; so the interaction of [H] and [p] in the confined law
lies between zero and D_gamma(H,p).
Orbit projection. If t = B_0 h in a representative, the orbit
receives the factor
w_gamma(t) = sum_{k in Z} exp( -(gamma/2)(t + 2 pi k)^2 ).
Poisson summation gives the uniform bound
sup_t | sqrt(2 pi gamma) w_gamma(t) - 1 |
<= 2 sum_{j>=1} e^{-j^2/(2 gamma)} =: delta_gamma -> 0
(gamma -> 0).
For gamma small enough that delta_gamma < 1, the normalized
projected density relative to the free orbit law lies between
(1-delta_gamma)/(1+delta_gamma) and its reciprocal. This
multiplicative estimate applies to the individual pin events and
their intersection, however small their probabilities, so their log
interactions differ by O(delta_gamma) uniformly in mesh and annulus.
Given a tolerance, first choose gamma so the orbit error is small,
then lambda large so the limiting absolute-pin interaction is small,
then take the mesh limit at that fixed geometry. This proves the
proposition. Nonnegativity in the free law follows by centered
precision comparison, or by sending gamma to zero at each fixed
finite graph. []
============================================================
7. GLUING KERNELS AND STABILITY
============================================================
Work with bare height weights, no confinement. The flat-pin
estimate gives an average bound for restoring an interface between
annuli; a uniform upper bound for the same kernel makes it stable
through any number of annuli.
Guard data and kernels. Use P, the guard and the end regions of
Section 6. In lattice units an annular block has walls R P and
lambda R P, with R large and dyadically aligned so all divisions
lie between rows of dual vertices; every block is connected, and
inter-block bonds are exactly the bonds to be restored.
Choose one guard vertex per annulus. Its guard datum u is the list
of all height differences from that vertex to the other guard
vertices in the chosen gauge. Every compatible datum has positive
probability. The guard law in block i and sector s is m_i^s(du).
Fixing the representative with chosen height zero turns u into full
prescribed data on the guard; the two sides are then conditionally
independent because the guard is a full separating band and the
energy is nearest-neighbour.
For two adjacent annuli with data u_i, u_{i+1} define
T_i^s(u_i,u_{i+1}) = Z_restored^s(u_i,u_{i+1})
/ Z_cut^s(u_i,u_{i+1}),
numerator summed modulo one common translation, denominator a
product of sums modulo one translation per block, so the numerator
sums the relative integer translation of the two blocks. The same
definition applies to a core (connected block filling the inner
hole of the first annulus, possibly with a magnetic source, no
prescribed guard) giving a kernel T_0^s(u_1), and to one connected
exterior attached to several disjoint last annuli (no direct bonds
between them) giving one joint kernel. In every case d_i = T_i^0(0),
0 meaning zero guard differences in the centered sector; no size
estimate for this positive scalar is needed.
LEMMA 7.1 (upper bound and centered average). For each kernel, every
sector and every compatible guard datum,
0 < T_i^s/d_i <= 1. (7.1)
For an interface between two annuli, choose on each side the end
region H containing its seam endpoints; for the joint exterior
kernel choose the outer end region of each incident annulus. Then
1 >= integral (T_i^0/d_i) prod_{incident l} dm_l^0
>= prod_{incident l} e^{-Dbar_l(H,p)}. (7.2)
Proof. Lemma 4.2 gives Z_restored^s(u)/Z_restored^0(0) <=
Z_cut^s(u)/Z_cut^0(0); rearranging gives (7.1). The comparison
applies to relative guard data (affine constraints on differences).
If auxiliary confinement is used, its divergent translation scalar
cancels within each displayed ratio before confinement is removed.
The two graphs need not have the same number of free components.
For the lower bound, work centered and impose [H] in every
indicated annulus. Adding these centered constraints to the
restored data-versus-zero ratio decreases it. Once they hold, choose
each annulus representative zero on its flat end; restoration is
then independent of guard data, the remaining variables being
relative integer translations and, for the exterior, unprescribed
exterior heights. With P_{l,0} the free centered law in annulus l,
comparison after division by the cut data-versus-zero ratio gives
T_i^0(u)/d_i >= prod_l P_{l,0}([H]|u_l) / P_{l,0}([H]|[p]).
All seam endpoints lie in the chosen end regions, so no other
annular heights enter restoration with these events imposed.
Averaging over the independent cut-block laws gives (7.2). []
LEMMA 7.2 (exact multiplication). Let a core be surrounded by N
annular blocks with no other adjacencies. Then
Z_full^s = Z_core^s prod_{i=1}^N Z_i^s
integral T_0^s(u_1)
prod_{i=1}^{N-1} T_i^s(u_i,u_{i+1})
prod_{i=1}^N dm_i^s(u_i). (7.3)
For several disjoint nests attached to a common connected exterior,
the formula includes its cut partition function and the one joint
exterior kernel.
Proof. Begin with one representative per cut block. On the block
tree, integer block constants modulo one common constant are in
bijection with integer differences along tree edges, with no
multiplicity factor (fix the root, recover others along unique
paths), so relative translations sum independently at interfaces.
Condition next on every guard datum: within an annulus inner and
outer configurations are conditionally independent, so the two
incident interface factors integrate separately. For several
nests, place the exterior at the root; its different boundary
portions need not be independent, so their integrations remain in
the single exterior kernel. This is exactly (7.3). []
SMALL AVERAGE LOSSES IN UNIT SECTORS. Call an annulus-annulus
interface regular when the connection is flat on the restored union
and has the same period 2 pi, -2 pi, or zero in both annuli
(restoring introduces no magnetic source).
PROPOSITION 7.3 (average loss). For a regular interface,
integral (T_i^s/d_i) prod_l dm_l^s
= ( integral (T_i^0/d_i) prod_l dm_l^0 )
* F_restored^s / F_cut^s. (7.4)
Given xi > 0, first choose eps small and fix P satisfying its disk
inclusions, then lambda large. There is a fixed aligned threshold
R_0 such that every regular interface with smaller inner scale at
least R_0 satisfies
integral (1 - T_i^s/d_i) prod_l dm_l^s <= xi (7.5)
in each of the two unit-period sectors and in the centered sector.
The individual flat-pin interactions in (7.2) can simultaneously be
made arbitrarily small.
Proof. The integral of T_i^s under the independent cut-block laws
is Z_restored^s/Z_cut^s; dividing this identity by its centered
counterpart gives (7.4). The same identity holds for a joint
exterior kernel, without yet a claim about its average loss.
For a circular annulus of radii 1 and lambda the minimum energy with
unit period 2 pi is 2 pi log lambda, attained by the angular
function; integration on concentric circles proves the matching
lower bound. Restriction of admissible forms, with
B(0,1-eps) subset P subset B(0,1+eps), gives
M(lambda P \ P^o) = 2 pi log lambda + O(eps),
constant independent of large lambda: the circular annulus with
radii 1+eps and lambda(1-eps) is contained in the polygonal one,
which is contained in the circular annulus with radii 1-eps and
lambda(1+eps). The same bounds with lambda^2 hold for the union of
two blocks, so
0 <= M(lambda^2 P \ P^o) - 2 M(lambda P \ P^o) <= C eps
(the lower bound by restriction to the two cut annuli, whose
energies agree by scale invariance in dimension two). Theorem 5.1,
applied at this fixed geometry, then makes the last factor in (7.4)
converge to a number in [ e^{-C eps/(2a)}, 1 ]. Changing the unit
period's sign does not change the energy.
Choose eps small for this error, then lambda large so Proposition
6.1 makes the two centered pin interactions small. Equations (7.2)
and (7.4) give the small loss in the mesh limit. With P and lambda
fixed, every regular interface is a refinement of one fixed
geometry; the convergence theorems hold along every admissible mesh
sequence, so one threshold R_0 works at every aligned scale. There
are only three sectors to consider, proving (7.5). []
A POSITIVE-KERNEL STABILITY LEMMA.
LEMMA 7.4 (stability from a positive initial weight). Let (X_i,m_i),
i >= 1, be probability spaces and t_i : X_i x X_{i+1} -> (0,1]
measurable kernels with
integral (1 - t_i) d(m_i x m_{i+1}) <= xi, 0 < xi < 1/64.
Fix measurable g : X_1 -> (0,1], positive m_1-almost everywhere.
For each N give X_1 x ... x X_N density proportional to
g(u_1) prod_{i=1}^{N-1} t_i(u_i,u_{i+1})
with respect to prod_i m_i. For all sufficiently large N its
last-coordinate marginal has density at most 2 with respect to m_N.
No positive essential lower bound on g is required.
Proof. Define L_i f(v) = integral t_i(u,v) f(u) dm_i(u). Propagate
the constant density:
q_1 = 1, beta_i = integral L_i q_i dm_{i+1},
q_{i+1} = beta_i^{-1} L_i q_i.
Inductively q_i <= 2, because
beta_i = 1 - integral (1 - t_i) q_i d(m_i x m_{i+1}) >= 1 - 2 xi,
q_{i+1} <= M := (1 - 2 xi)^{-1} < 2.
The same argument with kernels reversed bounds by 2 any normalized
density obtained by propagating 1 backwards through finitely many
kernels.
Propagate g with these same scalars rather than renormalizing:
f_1 = g, f_{i+1} = beta_i^{-1} L_i f_i,
c_i = integral f_i dm_i, E_i = f_i - c_i q_i.
The scalar c_i is the ratio of the two integrals starting from g
and from 1; equivalently the expectation of g under a normalized
backwards density on X_1, which is at most 2. Choose delta > 0 with
m_1(g < delta) < 1/4; every such density assigns at least half its
mass to {g >= delta}, so c_i >= delta/2 for every i.
For mean-zero E replace t_i in L_i E by t_i - 1. Since
0 <= (1 - t_i)^2 <= 1 - t_i, the resulting operator has L^2(m_i)-to-
L^2(m_{i+1}) norm at most sqrt(xi). The exact recurrence is
E_{i+1} = ( L_i E_i - ( integral L_i E_i dm_{i+1} ) q_{i+1} )
/ beta_i.
Using |q_{i+1}|_2 <= 2,
|E_{i+1}|_2 <= ( 3 sqrt(xi) / (1 - 2 xi) ) |E_i|_2, (7.6)
a strict contraction. Also t_i <= 1 gives |L_i E_i|_inf <= |E_i|_2,
so |E_{i+1}|_inf <= (3/(1-2xi)) |E_i|_2; the remainders tend to zero
in that norm. The last-coordinate density is f_N/c_N = q_N +
E_N/c_N, and c_N >= delta/2 with M < 2, so it is eventually bounded
by 2. []
APPLICATION TO A NEST OF ANNULI. Fix the parameters of
Proposition 7.3 with xi < 1/64 and increase R_0 if necessary. Set
R_j = R_0 lambda^j, j >= 0. Let annulus i have walls R_{i-1} P and
R_i P and fill the inner hole by the fixed core in R_0 P (a fixed
finite graph in lattice units, containing the source in a unit
sector; all regular annuli carry its period). Fix the core and the
connection on the increasing family of blocks once and for all, so
restrictions agree as N increases (a fixed straight connection cut
from the source). Thus the first guard law and the core interface
kernel do not vary with N.
COROLLARY 7.5 (last guard law). For these fixed choices the
last-guard law in a nest of N annuli has density at most 2 relative
to m_N^s for all sufficiently large N, in the centered sector and
either unit-period sector. The conclusion holds for identical
lattice translates of the nest with the same threshold in N.
Proof. Put t_i = T_i^s/d_i for regular interfaces and g = T_0^s/d_0
for the core interface. Lemma 7.1 gives 0 < g <= 1 and 0 < t_i <= 1;
Proposition 7.3 gives the average-loss hypothesis of Lemma 7.4;
Lemma 7.2 identifies the last marginal with the actual last-guard
law. The fixed initial weight g may be arbitrarily small on some
data; positivity suffices. A lattice translation preserves all
block laws and kernels; changing the source sign negates the
connection and heights; a connection with the same source restricts
to the chosen one up to an integral gauge change, which merely
reindexes guard data. Taking the maximum of the finitely many
thresholds for the centered and two unit sectors proves the
assertion. []
============================================================
8. SEPARATED CENTER-NORMALIZED CORRELATIONS
============================================================
Throughout this section k >= 1 is fixed, sigma_i in {-1,1}, and
x_{i,n} in D_n^o, x_{i,n} -> z_i in D,
z_i != z_j (i != j).
Write G_D(z,w) = (1/(2 pi)) log(1/|z-w|) + R_D(z,w), R_D smooth
across the diagonal on compact subsets of D x D, and define
H_D(z,sigma) = sum_{i=1}^k R_D(z_i,z_i)
+ 2 sum_{i<j} sigma_i sigma_j G_D(z_i,z_j).
No assumption on sum_i sigma_i is imposed.
LEMMA 8.1 (exterior energy). Remove closed disks of radius rho about
the distinct points z_1,...,z_k from D, with rho small enough that
they are disjoint and inside D. For affine gradients with periods
2 pi sigma_i about the holes, the minimum energy M of Theorem 5.1
satisfies
M = 2 pi k log(1/rho) + 4 pi^2 H_D(z,sigma) + O(rho). (8.1)
The error constant may depend on the separated interior
configuration, uniformly for small rho.
Proof. Set U(y) = sum_i sigma_i G_D(y,z_i) and rotate -2 pi nabla U
by a right angle, orientation chosen for the stated periods; the
result v is a smooth divergence-free affine gradient on the
punctured square, tangent to the outer boundary because U has zero
Dirichlet trace, hence admissible. Near z_i write
U(y) = (sigma_i/(2 pi)) log(1/|y-z_i|) + u_i(y),
u_i(z_i) = sigma_i R_D(z_i,z_i) + sum_{j!=i} sigma_j G_D(z_i,z_j),
u_i smooth and harmonic on fixed disjoint neighbourhoods. On a hole
boundary the radial singularity contributes no normal component to
v; the remaining normal component is bounded uniformly in rho and
has integral zero (a tangential derivative of u_i). For g in H^1 on
the punctured square, integration by parts with subtraction of the
mean of g on each inner circle gives
| integral v . nabla g | <= C rho |nabla g|_2 :
the L^2 norm of v . nu on such a circle is O(rho^{1/2}), while trace
and Poincare on the adjacent annulus of radii rho and 2 rho give
|g - circle mean|_{L^2(dB(z_i,rho))}
<= C rho^{1/2} |nabla g|_{L^2(B(z_i,2rho)\B(z_i,rho))},
collars disjoint. So minimizing |v + nabla g|_2^2 changes |v|_2^2 by
at most O(rho^2). Compute |v|_2^2 = (2 pi)^2 integral |nabla U|^2.
The outer-boundary contribution to Green's identity vanishes. On the
circle about z_i with nu into the hole, the integral of U d_nu U is
(1/(2 pi)) log(1/rho) + sigma_i u_i(z_i) + O(rho),
and the normal derivative of the regular harmonic part integrates to
zero (no logarithmically enhanced error). Summing over circles gives
(8.1); integrations may also be done with weak Dirichlet data on the
outer square, or by reflection across sides and corners away from the
sources. []
Apply to the rectilinear holes of the lattice decomposition. Let P
be the polygon of Section 6 with B(0,1-eps) subset P subset
B(0,1+eps); radii here are in scaled coordinates of D.
LEMMA 8.2 (moving lattice holes). Suppose rho_n -> rho > 0, rho
sufficiently small. Around x_{i,n} cut out the interiors in the
array division with hole shape rho_n P, using the rounding
conventions of Section 5. Let F_ext^s be the sector ratio in the
remaining dual graph with periods 2 pi sigma_i about these holes.
Then
limsup_n | log F_ext^s
+ ( 2 pi k log(1/rho) + 4 pi^2 H_D(z,sigma) ) / (2 a) |
<= C (eps + rho). (8.2)
C depends on the separated limiting configuration but not on small
rho or eps. The same holds after bounded-step mesh rounding.
Proof. Bracket the moving holes by fixed, positively separated
rectilinear holes with dyadic coordinates: start with circles of
radii (1 - 3 eps) rho and (1 + 3 eps) rho about the limiting points
and approximate them and their centers by still smaller dyadic
perturbations, chosen after fixing rho and eps. For all large n the
rounded moving holes lie between these fixed approximations and
contain their sources; the outer boundary is the same square. Use
one connection before any of these cuts: its restrictions to the
three exteriors have the same winding data, so the cut comparison
(4.1) brackets their sector ratios. Theorem 5.1 applies to each
fixed polygonal exterior, whose continuum minimum energy is bracketed
by the energies for the corresponding circular holes (restriction of
an admissible affine gradient preserves remaining periods). Changing
each radius by 1+O(eps) changes the logarithmic term of (8.1) by
O(eps), giving (8.2). Every domain here has positive passages and
separated hole boundaries; coherent rounding preserves connectivity
and the cycles defining periods. Only fixed-domain limits are used;
no uniformity under arbitrary motion of lattice marks. []
IDENTICAL NESTS AND EXTERIOR ATTACHMENT. Fix the annular parameters
eps, lambda, R_0 of Section 7 and write
R_j = R_0 lambda^j, N = N(n) = max{ j : R_j <= eta n },
eta > 0 small, the R_j now lattice radii. Place identical translated
nests (filled cores through N-th annuli) around all x_{i,n}. For
small eta these nests have disjoint closures, lie in the spin square,
and leave a connected exterior. Use the same parameters and the same
N for a single nest centered at 0, which represents a_n. Since
n x_{i,n} in Z^2, all cores and annuli are exact lattice translates;
restricted connections differ only by integral gauge changes, and
reversing a unit period leaves a bare nest ratio unchanged.
Pass to a subsequence with R_N/n -> t in [eta/lambda, eta]. For fixed
eta, lambda this is a positive macroscopic radius. The next estimate
concerns only the joint kernel attaching the last annuli to the
exterior, in the notation of Lemma 7.1: d_ext is the centered kernel
at zero relative guard data and m_{N,i}^s the cut law of the last
guard in the i-th annulus.
LEMMA 8.3 (exterior attachment). For every sufficiently small xi > 0
the parameters may be chosen so that (7.5) holds throughout the
regular annuli and
limsup_n integral ( 1 - T_ext^s/d_ext )
prod_{i=1}^k dm_{N,i}^s <= xi. (8.3)
This holds for shifted and centered calculations and for the single
center nest. Parameters depend on the fixed separated configuration
and on xi, not on n. Errors in (8.2) at the radii of this
construction may simultaneously be made at most xi.
Proof. Centered: (7.2) bounds the attachment loss by the sum of the
flat-pin interactions in the incident last annuli, made small by the
choices of Section 6; the bound depends only on these annuli.
Shifted: use the exact identity (7.4) for the last annuli with their
common exterior. Restoring produces an exterior whose smaller holes
have radius parameter tending to t/lambda. Before restoration the
factors are the k last-annulus sector ratios and the exterior ratio
with holes of radius parameter t. By Theorem 5.1 each annular energy
is 2 pi log lambda + O(eps); by (8.2) restored and cut exterior
energies have the same term 4 pi^2 H_D(z,sigma). The difference of
radius terms is exactly k 2 pi log lambda, so the logarithm of the
restored ratio divided by the product of cut ratios has absolute
limiting error at most C(eps + eta). Circular brackets make this
uniform over t in [eta/lambda, eta] and over large fixed lambda.
Choose eps and eta small, then lambda large for the flat-pin
interactions, then R_0 large for the fixed annular geometry, keeping
(7.5). The centered estimate, the restored-to-cut identity and
T_ext^s/d_ext <= 1 now give (8.3) after making preliminary errors
small enough in terms of xi; apply the same choices to the
one-source configuration at 0. The argument was given after
extracting a limit t; every subsequence has such a further
subsequence with uniform error bounds, so it proves the stated
limiting bounds on the full sequence. All height and pin limits
precede variation of geometric parameters; no rate uniform in a
shrinking physical radius is assumed. []
PROPOSITION 8.4 (center-normalized correlations). For the separated
moving marks fixed at the start of the section,
lim_n log( E_n^0 exp( i sum_i sigma_i theta_{x_{i,n}} ) / a_n^k )
= -( H_D(z,sigma) - k R_D(0,0) ) / (2 K_*). (8.4)
Proof. Let F_nest(R_N) be the bare sector ratio of one whole free
nest with a unit source, including its core; it is the same for all
translated sources, either sign, and the center reference. By Lemma
7.2 the partition function of the whole graph is the product of cut
block partition functions times the gluing-kernel expectation. After
the internal kernels are integrated, the last-guard law has density
at most 2 relative to m_N^s for large N, by Corollary 7.5.
Let e_s be the expectation of T_ext^s/d_ext under the product of
propagated last-guard laws, e_0 the centered analogue. Equation
(8.3) with the product density bound gives
1 - 2^k xi - o(1) <= e_s <= 1 in both sectors. The centered factor
d_ext cancels between numerator and denominator, leaving
E_n^0 exp( i sum_i sigma_i theta_{x_{i,n}} )
= F_nest(R_N)^k F_ext^s (e_s/e_0).
Take logarithms with xi small at fixed k:
log E_n^0 exp( i sum_i sigma_i theta_{x_{i,n}} )
= k log F_nest(R_N) + log F_ext^s + O_k(xi) + o(1).
The same calculation with the single center source expresses log a_n
using this identical nest ratio; subtracting k times it cancels the
microscopic term exactly. In (8.2) the common-radius logarithms also
cancel, leaving coefficient 4 pi^2/(2 a) = 1/(2 K_*) since a = 8 pi
and K_* = 2/pi. This proves (8.4) up to an error made arbitrarily
small. The nest choices may depend on that accuracy; harmless, since
the left side is a fixed exact observable. If convergence failed, a
subsequence with fixed positive error, at high enough chosen
accuracy, would have a further subsequence with R_N/n -> t and
smaller error by the preceding calculation, a contradiction. []
THE DETERMINISTIC DIAGONAL CORRECTION.
LEMMA 8.5 (Green diagonal differences). If x_n in D_n^o and
x_n -> z in D, then
G_n(x_n,x_n) - G_n(0,0) -> R_D(z,z) - R_D(0,0), (8.5)
uniformly bounded when x_n ranges over a fixed compact subset of D.
Proof. The killed Green estimate (companion Bessel paper, Lemma 4.2)
gives local uniform off-diagonal convergence of G_n to G_D with the
same unscaled Laplacian convention. We recover the difference of
diagonals without specifying their common lattice constant. Choose a
small fixed box Q of displacements about 0 with x_n + Q and Q in D
for large n. For lattice vectors v set h_n(v) = G_n(x_n,x_n+v) -
G_n(0,v); both kernels have the same unit source at v = 0, so h_n is
discrete harmonic throughout Q, including at v = 0. On a coherently
rounded boundary of Q, off-diagonal convergence identifies its limit
as G_D(z,z+v) - G_D(0,v) = R_D(z,z+v) - R_D(0,v). The discrete
maximum principle bounds h_n(0) between the min and max of these
boundary values up to an error -> 0. Shrinking Q and continuity give
(8.5). If uniform boundedness on an interior compact failed, a
violating sequence would have a convergent subsequence of locations,
contradicting (8.5); finitely many smaller lattice sizes cause no
difficulty. []
For sigma in {-1,1} define normalized point fields
W_n^sigma(x) = a_n^{-1}
exp( (G_n(x,x) - G_n(0,0)) / (2 K_*) ) e^{i sigma theta_x}.
Then S_n(f) = n^{-2} sum_x f(x) W_n^+(x), and changing the sign
conjugates the point field without conjugating the test function.
COROLLARY 8.6 (normalized separated correlations). For every fixed
separated moving configuration,
lim_n E_n^0 prod_{i=1}^k W_n^{sigma_i}(x_{i,n})
= exp( -(1/K_*) sum_{i<j} sigma_i sigma_j G_D(z_i,z_j) ). (8.6)
The convergence to the same kernel at the lattice locations is
uniform when those locations range over a compact interior set with
mutual distances bounded below by a fixed positive number.
Proof. Exponentiate (8.4), multiply by the deterministic factors and
use (8.5); the terms sum_i R_D(z_i,z_i) - k R_D(0,0) cancel
exactly. Uniformity: a sequence of violations in compact separated
configuration space would have a convergent subsequence, to which
the sequential statement applies. []
============================================================
9. A UNIFORM TWO-POINT BOUND NEAR THE DIAGONAL
============================================================
The separated formula alone does not control moment sums near
collisions. A weaker integrable power bound suffices, obtained by
cutting two disjoint nests at a scale comparable to the distance
between marks.
Fix the annular parameters once and for all, with accuracy high
enough that the center attachment of Lemma 8.3 is bounded away from
zero, and keep N(n) = max{ j : R_j <= eta n }.
LEMMA 9.1 (nest lower bounds). There are c, c' > 0, independent of
n, N, j, such that for large n
a_n >= c F_nest(R_{N(n)}), (9.1)
and for every N >= j >= 0
F_nest(R_N) >= c' lambda^{-(N-j)/2} F_nest(R_j), (9.2)
where j = 0 means the filled core alone.
Proof. For the single center source, the normalized exterior
attachment has a positive lower bound in the shifted sector and is at
most one in the centered sector; the exterior sector ratio is bounded
away from zero by (8.2), since every subsequential hole radius lies
in [eta/lambda, eta]. The exact gluing identity gives (9.1), with the
compact subsequence argument making the bound uniform for large n.
Add one regular annulus to a nest. Its sector ratio multiplies the
nest ratio, together with the ratio of shifted and centered
attachment expectations, normalized by their common centered
zero-data value. The centered expectation is at most one; once the
nest is long enough the last-data density is at most two, so (7.5)
makes the shifted expectation at least 1 - 2 xi. By Theorem 5.1 and
the circular annular energy bracket, the annular sector factor at
large scales is bounded below by
lambda^{-1/8} exp( -O(eps) - o(1) ), pi/a = 1/8.
Include in the fixed parameter choices that lambda and then the
lattice threshold are large. Each sufficiently late extension factor
is at least lambda^{-1/2}; finitely many earlier extension factors
are strictly positive, absorbable into c' > 0 valid for every
interval of extension indices. Multiplication proves (9.2). []
PROPOSITION 9.2 (two-point collision bound). For every compact E in
D there is C_E < infinity with, for all n >= 1, all x,y in E and
D_n^o (including x = y) and all sigma, tau in {-1,1},
0 <= E_n^0[ W_n^sigma(x) W_n^tau(y) ]
<= C_E ( |x-y| + 1/n )^{-1}. (9.3)
Expectations of products of any number of the point fields W_n^sigma,
with arbitrary signs and repeated locations, are also nonnegative.
Proof. Write r = n|x-y| and choose c_E > 0 so small that nests of
radius R <= c_E r centered at x and y (lattice units) are disjoint
and inside the spin square; decrease c_E so c_E r <= eta n for all
locations considered.
If there is a largest j >= 0 with R_j <= c_E r, cut out these two
nests. The restriction of the magnetic sector to each nest has a
single unit source. The cut comparison and the bound one for the
remaining sector ratio give
E_n^0 e^{i sigma theta_x + i tau theta_y}
<= F_nest(R_j)^2.
Equations (9.1) and (9.2) give
E_n^0 e^{i sigma theta_x + i tau theta_y} / a_n^2
<= C lambda^{N-j}.
Maximality of j gives R_j > c_E r / lambda, while R_N <= eta n;
since R_N/R_j = lambda^{N-j}, this is at most C_E n/(1+r), increasing
the constant.
If no such j exists then r < R_0/c_E. Use the bare bound by one and
(9.2) with j = 0 together with (9.1): the fixed core ratio is
positive, so
a_n^{-2} <= C lambda^N <= C n <= C_E n/(1+r),
covering repeated locations with either pair of signs. The
deterministic Green factors in the two W_n are uniformly bounded on
E by Lemma 8.5 and preserve the bound; increasing C_E handles
finitely many smaller lattice sizes.
Nonnegativity: the Fourier current expansion of the pinned XY law
expresses each integer-charge correlation as a sum of nonnegative
weights divided by a positive partition function. The fixed boundary
imposes no neutrality constraint; repeated locations add their
charges. Multiplication by the positive deterministic factors proves
all asserted nonnegativity. []
============================================================
10. THE CONTINUUM FIELD AND THE FIELD LIMIT
============================================================
Section 8 gives signed correlations at separated points and
Proposition 9.2 a bound up to the two-point diagonal. We remove
collisions of arbitrarily many sites using only the two-point bound,
aided by a Lee-Yang absolute-moment estimate, then construct the
continuum field and identify its moments and law.
Throughout, S_n^sigma(f) = n^{-2} sum_{x in D_n^o} f(x) W_n^sigma(x),
sigma in {-1,1}, with W_n^sigma as in Section 8; thus S_n^+ = S_n and
S_n^-(f) = overline{S_n^+(overline f)}. The same convention is used
for continuum fields: the minus sign conjugates the field, and
conjugating an evaluation conjugates its test function.
A. Absolute moments from Lee-Yang.
LEMMA 10.1 (Lee-Yang absolute moment bound). Consider a finite
ferromagnetic XY model with nonnegative edge couplings and all
boundary angles fixed to zero. For deterministic weights lambda_x >=
0 set W = sum_x lambda_x e^{i theta_x}, v = E|W|^2. There is an
absolute constant C with
|W|_{L^p} <= C sqrt(p) v^{1/2}, p >= 2. (10.1)
For every finite list of vertices, including repetitions, and every
choice of signs sigma_i in {-1,1},
E exp( i sum_i sigma_i theta_{x_i} ) >= 0. (10.2)
Proof. Let U be independent uniform on the circle and rotate every
spin, boundary included, by U; subtracting the common boundary angle
recovers the original fixed-boundary law. Put X = Re(e^{iU} W). The
finite-graph XY Lee-Yang theorem with the common uniformly
distributed boundary convention says E e^{zX} has only imaginary
zeros (Newman-Wu 2019, Theorem 1 and Remark 5). That boundary
convention on simple graphs follows by adjoining a vertex of
observable weight zero, joining it ferromagnetically to every
boundary vertex and taking the added couplings to infinity; limiting
boundary spins equal the new spin, uniform by rotation invariance,
and the bounded observable gives locally uniform convergence of
Laplace transforms, so zero-free half-planes persist.
X is symmetric and bounded, so its Lee-Yang product representation
(Newman-Wu 2019, Proposition 13) is
E e^{zX} = e^{B z^2} prod_j (1 + z^2/y_j^2),
with B >= 0, sum_j y_j^{-2} < inf, B + sum_j y_j^{-2} = (1/2) E X^2,
zeros listed as pairs +-i y_j. Since uniform rotation gives
E X^2 = v/2, the inequality 1+u <= e^u yields
E e^{tX} <= e^{t^2 v/4}, t in R. (10.3)
Y = Im(e^{iU} W) has the same law; for v > 0, Chernoff applied to
both signs of X and Y gives
P( |W| > r ) <= 4 e^{-r^2/(2v)}, r > 0.
Integrating against p r^{p-1} dr gives E|W|^p <= 2p(2v)^{p/2}
Gamma(p/2), hence (10.1) (v = 0 immediate; |W| unchanged by the
rotation).
For (10.2), orient edges arbitrarily and expand each interaction in
its absolutely convergent Fourier series. For coupling b >= 0,
I_k(b) = sum_{r>=0} (b/2)^{2r+|k|} / ( r! (r+|k|)! ) >= 0,
k in Z,
(interpreted as I_0(0) = 1, I_k(0) = 0 for k != 0). Integrating each
interior angle imposes the divergence constraint on integer edge
currents prescribed by inserted charges; boundary angles are zero,
contributing a factor one and imposing no constraint. Each surviving
summand is nonnegative and the partition function is positive;
repeated vertices add their charges. []
B. The continuum circle field.
LEMMA 10.2 (circle Wick field and mixed moments). Let K > 1/(4 pi)
and let Phi_D be the zero-Dirichlet GFF with covariance G_D. Where
the circle of radius eps about x lies in D, define
V_eps(x) = exp( i Phi_{D,eps}(x) / sqrt(K)
+ E Phi_{D,eps}(x)^2 / (2 K) ).
There is a complex random distribution V_{K,D} in H^{-2}_loc(D) such
that for every f in C_c^inf(D;C), V_eps(f) -> V_{K,D}(f) in every
finite L^p; for every smooth compactly supported cutoff chi,
chi V_eps -> chi V_{K,D} in L^2(Omega;H^{-2}). For m >= 1, signs
sigma_i, and f_i in C_c^inf(D;C),
E prod_{i=1}^m V_{K,D}^{sigma_i}(f_i)
= integral_{D^m} prod_i f_i(x_i)
exp( -(1/K) sum_{i<j} sigma_i sigma_j G_D(x_i,x_j) )
dx_1 ... dx_m, (10.4)
absolutely convergent. In particular E V_{K,D}(f) = integral_D f.
Proof. Fix compact E in D containing the test supports and 0 < rho <
1 with its closed 2 rho-neighbourhood in D. Write
G_D(u,v) = (1/(2 pi)) log(1/|u-v|) + R_D(u,v),
q = 1/(2 pi K) < 2,
R_D smooth and bounded across the diagonal. Let C_{a,b}(x,y) be the
covariance of circle averages of Phi_D/sqrt(K) with centers x,y in E
and radii a,b <= rho. First, uniformly in both radii,
C_{a,b}(x,y) = q log(1/max{|x-y|,a,b}) + O_{E,K}(1). (10.5)
Put r = |x-y|, M = max{r,a,b}. The double circle average of the
log-distance equals
L_{a,b}(r) = (1/(2 pi)) integral_0^{2 pi}
log max{a, |r + b e^{it}|} dt,
using the identity (1/(2 pi)) integral_0^{2 pi} log|z - a e^{it}| dt
= log max{|z|, a}. The integrand is at most log(2M). If a = M it is
at least log M; otherwise its integral is at least the circle mean of
log|r + b e^{it}| = log max{r,b} = log M. Hence log M <= L_{a,b}(r)
<= log M + log 2, proving (10.5) with the bounded remainder, even for
coincident centers; finite logarithmic integrals justify the
covariances.
For radii eps_i <= rho, Gaussian integration with the full diagonal
variances cancelled gives
E prod_i V_{eps_i}^{sigma_i}(x_i)
= exp( - sum_{i<j} sigma_i sigma_j
C_{eps_i,eps_j}(x_i,x_j) ). (10.6)
To dominate this for all signs and collision patterns, take m >= 2,
distinct centers, and set
d_i = min{ rho, min_{j!=i} |x_i - x_j| },
r_i = max{ eps_i, d_i/4 }.
All circles with radii r_i stay in D, and for i != j,
max{ |x_i-x_j|, r_i, r_j } = max{ |x_i-x_j|, eps_i, eps_j }, so
enlarging radii changes each off-diagonal covariance by a bounded
amount. If C^r is the enlarged covariance matrix, positive
semidefiniteness gives
- sum_{i<j} sigma_i sigma_j C^r_{ij}
= (1/2) sum_i C^r_{ii} - (1/2) sum_{i,j} sigma_i sigma_j C^r_{ij}
<= (1/2) sum_i C^r_{ii}.
Since C^r_{ii} <= q log(4/d_i) + C_{E,K},
0 < E prod_i V_{eps_i}^{sigma_i}(x_i)
<= C_{E,K,m} prod_i d_i^{-q/2}, (10.7)
independent of all radii including their ratios; for m = 1 the
expectation is one.
Integrability on E^m. Put alpha = q/2 < 1 and use
d_i^{-alpha} <= rho^{-alpha} + sum_{j!=i} |x_i-x_j|^{-alpha}.
Expanding the product produces finitely many directed graphs on
{1,...,m} with at most one outgoing edge per vertex and no
self-loop; an edge i -> j contributes |x_i-x_j|^{-alpha}. Each
component is a tree or one directed cycle with trees attached.
Vertices with no incoming edge integrate first, each with at most one
remaining incident edge, using
sup_{y in E} integral_E |x-y|^{-s} dx < inf, 0 <= s < 2, (10.8)
by polar integration over a disk containing E - E. This removes all
trees at uniform cost. A two-cycle contributes |x-y|^{-2 alpha},
integrable since 2 alpha = q < 2. A cycle on at least three vertices:
integrate one vertex by Cauchy-Schwarz,
integral_E |x-y|^{-alpha} |x-z|^{-alpha} dx
<= ( integral_E |x-y|^{-2 alpha} dx )^{1/2}
( integral_E |x-z|^{-2 alpha} dx )^{1/2}
<= C_{E,alpha},
leaving a path integrable from its ends; isolated vertices contribute
|E|. Thus every term is integrable, including simultaneous
collisions of any number of centers.
At distinct centers C_{eps_i,eps_j}(x_i,x_j) -> G_D(x_i,x_j)/K as
all radii -> 0, with no restriction on relative rates. Equations
(10.6) and (10.7) give convergence of all integrated mixed kernels by
dominated convergence; the second mixed kernels with radii eps, delta
show V_eps(f) is L^2-Cauchy. Equal positive and negative signs give
uniform bounds for every even absolute moment, and for any finite p
choose a larger even order; these bounds plus L^2 convergence give
L^p convergence by uniform integrability. Holder passes every mixed
product to the limit, proving (10.4).
Realize the limits as one distribution. Choose a flat torus whose
fundamental square contains closure(D) in its interior with
normalized Fourier basis e_nu, nu in Z^2. For fixed chi in
C_c^inf(D), (10.7) with two factors gives
sup_{nu, 0<eps<eps_chi} E|V_eps(chi e_nu)|^2 < inf
for small eps_chi > 0. Each Fourier coefficient is L^2-Cauchy and
sum_nu (1+|nu|^2)^{-2} < inf; dominated convergence in this series
shows chi V_eps is L^2(Omega;H^{-2})-Cauchy on the torus. Choose a
countable exhaustion by smooth cutoffs equal to one near the supports
of their predecessors; their limits are compatible since smooth
multiplication is continuous on H^{-2}. They define an a.s.
H^{-2}_loc distribution V_{K,D}, and testing recovers the scalar
limits already built. Localization identifies torus and Euclidean
local Sobolev spaces on D. The normalization uses the complete
Dirichlet variance, including its finite spatial part. []
At K = K_* = 2/pi the exponent q = 1/4, and Lemma 10.2 constructs
exactly V_*. We now prove its mixed moments are the limits of the
lattice moments.
C. Uniform bounds and removal of collisions.
LEMMA 10.3 (bounds for smeared lattice fields). For every compact E
in D, every bounded complex f supported in E, and every p >= 2,
E_n^0 |S_n^sigma(f)|^2 <= C_E |f|_inf^2,
|S_n^sigma(f)|_{L^p} <= C_E sqrt(p) |f|_inf, (10.9)
uniformly in n and sigma. If B subset E is a square of side d <= 1,
E_n^0 |S_n^sigma(1_B)|^2 <= C_E (d + 1/n)^3,
|S_n^sigma(1_B)|_{L^p} <= C_E sqrt(p) (d + 1/n)^{3/2}. (10.10)
Proof. Write h = 1/n. For fixed x in B and D_n^o the shell
j h <= |x-y| < (j+1) h contains at most C(j+1) lattice points, so
(9.3) implies
h^2 sum_{y in B and D_n^o} (|x-y| + h)^{-1}
<= C h sum_{0 <= j <= C(d/h+1)} 1 <= C(d+h).
There are at most C(d/h+1)^2 choices of x, so a second summation
including its factor h^2 proves the first bound of (10.10).
Summation over a fixed compact set gives a uniform bound for its
indicator. Positivity of the opposite-sign two-point expectation
gives, for arbitrary complex f supported there,
E_n^0 |S_n^+(f)|^2
<= h^4 sum_{x,y in E and D_n^o} |f(x)| |f(y)|
E_n^0[ W_n^+(x) W_n^-(y) ]
<= C_E |f|_inf^2.
For a nonnegative weight w the coefficients of the spins in S_n^+(w)
are lambda_x = n^{-2} w(x) a_n^{-1}
exp( (G_n(x,x) - G_n(0,0))/(2 K_*) ) >= 0. Lemma 10.1 applies to
these exact coefficients for each n, giving the second bound of
(10.10) from the first. For complex f decompose
f = (Re f)_+ - (Re f)_- + i (Im f)_+ - i (Im f)_-;
each nonnegative summand is bounded by |f|_inf and supported in E.
Minkowski and the Lee-Yang bound prove the second estimate of
(10.9); conjugation gives the same bounds for the minus field. []
PROPOSITION 10.4 (convergence of every mixed moment). For every
m >= 1, signs sigma_i, and f_i in C_c^inf(D;C),
lim_n E_n^0 prod_{i=1}^m S_n^{sigma_i}(f_i)
= E prod_{i=1}^m V_*^{sigma_i}(f_i), (10.11)
the right side being (10.4) with K = K_*.
Proof. Let E in D contain all supports. The case m = 1 follows from
(8.6), uniformly on E, and Riemann summation. Assume m >= 2 and
expand the moment over (D_n^o)^m. All signed correlations in this sum
are nonnegative by (10.2) and positivity of the deterministic
normalizers.
Fix i != j and small d > 0. Enlarge the squares of a side-d grid by
a fixed factor to get a family B_d of at most C_E d^{-2} squares,
each of side at most C d, such that every pair x_i, x_j in E with
|x_i - x_j| < d lies in one of them; all lie in a fixed larger
compact E' in D when d is small. Positivity bounds the absolute
value of the restricted moment sum by
( prod_l |f_l|_inf ) sum_{B in B_d}
E_n^0[ S_n^{sigma_i}(1_B) S_n^{sigma_j}(1_B)
prod_{l != i,j} S_n^{sigma_l}(1_{E'}) ].
Each expectation is nonnegative, so at most the expectation of the
product of absolute values. Holder with m equal exponents, Lemma 10.1
and Lemma 10.3 bound it by
C_{E',m} |S_n^+(1_B)|_{L^2}^2 <= C_{E',m} (d + 1/n)^3.
Indeed the two box factors each contribute their L^2 norm times a
constant depending only on m, and every other factor has uniformly
bounded L^m norm. Summing over all pairs proves
limsup_n n^{-2m} sum_{x_1,...,x_m in D_n^o,
min_{i<j}|x_i-x_j| < d} ( prod_i |f_i(x_i)| )
E_n^0 prod_i W_n^{sigma_i}(x_i)
<= C_{E,m,f_1,...,f_m} d, (10.12)
which includes repeated lattice sites of every multiplicity and every
sign choice.
Choose a continuous cutoff on configurations that vanishes when some
pair is at distance at most d and equals one when every pair is at
distance at least 2d. On its support the uniform separated-point
convergence (8.6) turns the lattice moment sum into the Riemann sum
of the continuum kernel; the cutoff makes the product with the test
functions continuous across excluded diagonals, so for fixed d these
sums converge to their continuum integrals. Equation (10.12) bounds
the omitted lattice contribution by a quantity -> 0 with d after the
upper limit in n, and the omitted continuum contribution -> 0 by the
integrable domination (10.7). Let d -> 0 to get (10.11). []
D. Identification of laws and local tightness.
PROPOSITION 10.5 (joint convergence of test-function laws). For
every k >= 1 and f_1,...,f_k in C_c^inf(D;C),
( S_n(f_1),...,S_n(f_k) ) => ( V_*(f_1),...,V_*(f_k) ),
and every mixed moment of this vector and its complex conjugate
converges.
Proof. Fix a real linear combination Z_n of the real and imaginary
parts of the k lattice evaluations, and let Z be the same
combination of the continuum evaluations. Equation (10.9) gives
|Z_n|_{L^p} <= C sqrt(p) for all p >= 2, C depending on the tests and
the combination. Hence, for every t >= 0,
sup_n E_n^0 e^{t |Z_n|} < inf, (10.13)
since the exponential series is dominated, apart from its first two
terms, by sum_{p>=2} (C t sqrt(p))^p / p! < inf. Proposition 10.4
gives convergence of even moments of Z_n to those of Z; Z then obeys
the same bound at even orders, and at other orders by Holder. So Z
has every exponential moment, and absolute convergence of the
exponential series determines its characteristic function from its
moments.
The finite-dimensional lattice vectors are tight by their second
moments. Along a weakly convergent subsequence, the higher uniform
moment bounds make every polynomial in their real and imaginary
coordinates uniformly integrable, so the limit has the moments of
Proposition 10.4. Each of its real projections has the same even
moment bounds as Z, hence is exponentially integrable and has the
same characteristic function as Z. The Cramer-Wold criterion
identifies the vector law. Every convergent subsequence has this law,
proving full-sequence convergence. For the stated mixed moments use
overline{S_n(f)} = S_n^-(overline f) in Proposition 10.4. []
PROPOSITION 10.6 (tightness in the local Sobolev topology). The laws
of S_n are tight in H^{-3}_loc(D) and converge there along the full
sequence to the law of V_*.
Proof. Use the larger flat torus of Lemma 10.2 and its normalized
Fourier basis e_nu. Choose real smooth cutoffs chi_j with compact
support in D, equal to one on an increasing exhaustion of D and with
chi_{j+1} = 1 near supp chi_j. Since Fourier functions have
uniformly bounded suprema, (10.9) gives
sup_{n,nu} E_n^0 |S_n(chi_j e_nu)|^2 <= C_j.
Fourier summation yields
sup_n E_n^0 | chi_j S_n |_{H^{-2}}^2
<= C'_j sum_{nu in Z^2} (1 + |nu|^2)^{-2} < inf. (10.14)
The embedding H^{-2} -> H^{-3} on the torus is compact: finite Fourier
truncations have finite rank and on their complements the ratio of
squared Sobolev weights is (1+|nu|^2)^{-1} -> 0.
Given eta > 0 choose R_j < inf so the sum of Markov bounds from
(10.14) with thresholds R_j^2 is less than eta. With probability at
least 1 - eta all | chi_j S_n |_{H^{-2}} <= R_j hold. The set of
distributions satisfying these inequalities is relatively compact in
H^{-3}_loc(D): compact embedding gives a convergent subsequence for
each localized field, a diagonal choice converges for all j, and the
identities chi_j (chi_{j+1} S_n) = chi_j S_n pass to limits since
smooth multiplication is continuous on H^{-3}; the limits define one
distribution on D. This proves tightness.
The topology is the Euclidean local topology of Theorem 1.1: on
distributions supported in a fixed compact subset of D, torus and
Euclidean Sobolev norms are equivalent by localization inside the
fundamental square, and every compactly supported cutoff is supported
where some chi_j equals one; the localized norms generate exactly
H^{-3}_loc(D).
Every subsequential limit has the test-function laws of Proposition
10.5, since evaluation on a smooth compactly supported test is
continuous. These laws determine the distribution-valued law: the
tests chi_j e_nu determine every localized Fourier coefficient, and
their real and imaginary parts generate the Borel sigma-field of the
compatible subspace of the countable product of separable H^{-3}
spaces. Lemma 10.2 places V_* in this space, so every subsequential
limit has its law. Tightness and uniqueness give full-sequence
convergence. []
PROOF OF THEOREM 1.1. Propositions 10.4, 10.5 and 10.6 give the
moment convergence, the joint laws of smeared fields, and convergence
in law in H^{-3}_loc(D) to V_*. Every lattice field in the argument
uses the same exact center magnetization a_n and the same fixed
Green-function factor, and the microscopic nest ratio cancels
exactly against the center observable (Proposition 8.4). This
completes the proof. []
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REFERENCES
============================================================
[1] V. L. Berezinskii. Destruction of long-range order in
one- and two-dimensional systems with a continuous symmetry
group. I. Classical systems. Soviet Physics JETP 32 (1971),
493-500.
[2] J. Frohlich and T. Spencer. The Kosterlitz-Thouless transition
in two-dimensional Abelian spin systems and the Coulomb gas.
Comm. Math. Phys. 81 (1981), 527-602.
[3] J. Junnila, E. Saksman, C. Webb. Imaginary multiplicative chaos:
moments, regularity and connections to the Ising model. Ann.
Appl. Probab. 30 (2020), 2099-2164.
[4] J. M. Kosterlitz. The critical properties of the two-dimensional
XY model. J. Phys. C 7 (1974), 1046-1060.
[5] J. M. Kosterlitz and D. J. Thouless. Ordering, metastability and
phase transitions in two-dimensional systems. J. Phys. C 6
(1973), 1181-1203.
[6] P. Lammers. Bijecting the BKT transition. Preprint (2023),
arXiv:2301.06905.
[7] O. A. McBryan and T. Spencer. On the decay of correlations in
SO(n)-symmetric ferromagnets. Comm. Math. Phys. 53 (1977),
299-302.
[8] N. D. Mermin and H. Wagner. Absence of ferromagnetism or
antiferromagnetism in one- or two-dimensional isotropic
Heisenberg models. Phys. Rev. Lett. 17 (1966), 1133-1136.
[9] D. R. Nelson and J. M. Kosterlitz. Universal jump in the
superfluid density of two-dimensional superfluids. Phys. Rev.
Lett. 39 (1977), 1201-1205.
[10] C. M. Newman. Inequalities for Ising models and field theories
which obey the Lee-Yang theorem. Comm. Math. Phys. 41 (1975),
1-9.
[11] C. M. Newman and W. Wu. Gaussian fluctuations for the classical
XY model. Ann. Inst. Henri Poincare B 54 (2018), 1759-1777.
[12] C. M. Newman and W. Wu. Lee-Yang property and Gaussian
multiplicative chaos. Comm. Math. Phys. 369 (2019), 153-170.
[13] O. Regev and N. Stephens-Davidowitz. An inequality for Gaussians
on lattices. SIAM J. Discrete Math. 31 (2017), 749-757.
[14] D. van Engelenburg and M. Lis. An elementary proof of phase
transition in the planar XY model. Comm. Math. Phys. 399 (2023),
85-104.
[15] OpenAI. BKT universality for height and planar spin fields.
OpenAI Math Release preprint, 2026.
[16] OpenAI. The critical correlation exponent of the planar XY
model. OpenAI Math Release preprint, 2026.
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