Does a portion of dimer conguration determines its domain of denition Antoine Bannier Beno t Laslier

2025-05-03 0 0 642.83KB 30 页 10玖币
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Does a portion of dimer configuration determines its domain
of definition?
Antoine Bannier Benoˆıt Laslier
October 25, 2022
Abstract
Critical models are, almost by definition, supposed to feature both slow decay of corre-
lations for local observables while retaining some mixing even for macroscopic observables.
A strong version of the latter property is that changing boundary conditions cannot have a
singular (in the measure theoretic sense) effect on the model away from the boundary, even
asymptotically. In this paper we prove that statement for the wired uniform spanning tree
and temperleyan dimer model.
Contents
1 Introduction 2
2 Assumptions and background 4
2.1 Graphs and random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Loopsoup...................................... 7
2.3 Uniform spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Uniformisation and prime ends . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Dimers, winding, and height function . . . . . . . . . . . . . . . . . . . . . . . 12
3 Upper bound on dν1
dν214
3.1 Coupling....................................... 14
3.2 Radon-Nikodym bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Lower bound on dν1
dν216
4.1 Mixed Loop-erasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 RW in D2...................................... 18
4.3 RW in D1...................................... 19
4.4 loop-erasure..................................... 21
4.5 Coupling several paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Extensions and generalisations 23
5.1 Continuity with respect to the domains . . . . . . . . . . . . . . . . . . . . . 23
5.2 Dimer configurations and height functions . . . . . . . . . . . . . . . . . . . . 24
5.3 Samplinganannulus................................ 27
5.4 Continuumlimit .................................. 28
1
arXiv:2210.13398v1 [math.PR] 24 Oct 2022
1 Introduction
The goal of this paper is to establish strong mixing estimates for both the two dimensional
wired uniform spanning tree (UST from now on) and the dimer model. Focusing on the
former for concreteness and because it will be the main language used in the paper, we
wish to say that as soon as we look macroscopically away from the wired boundary, the law
of an UST “forgets” any small detail of the boundary and only keeps a “fuzzy memory”
of even the macroscopic location of that boundary. To give a precise sense to the above
sentence, we consider the following statistical setup. Fix U, D1, D2be three simply connected
bounded open sets, with U⊂D1∩D2and let U#δ, D#δ
1, D#δ
2be discretisations of these sets
(ultimately we will want to be as general as possible here but, for now, it is enough to think
that U#δ=U∩δZ2and similarly for the others). Our mixing statement then becomes a
testing problem: if one is given the restriction to U#δof some UST, can we say whether it
was sampled from D#δ
1or D#δ
2? By “fuzzy memory”, we mean that, even asymptotically
as δ→0, there is an upper bound on the quality any statistical test can hope to achieve.
More precisely our main result is the following.
Theorem 1.1. Let ν1, ν2denote the laws of the restrictions to Uof wired UST in D#δ
1and
D#δ
2respectively. For all  > 0, there exits C=C(U, D1, D2, )>0such that for all δsmall
enough
ν1(1
C≤dν1
dν2≤C)≥1−, ν2(1
C≤dν1
dν2≤C)≥1−.
Essentially one can think of that result as stating that ν1and ν2are mutually absolutely
continuous in a way which is independent of δ.
In fact Theorem 1.1 and the techniques used to prove it will provide a number of ex-
tensions which we hope can form a base set of mixing statements to be used as a toolkit
in later work. We defer to Section 5 for any details but briefly Corollary 5.1 contains the
aforementioned fact that microscopic details of the boundary are forgotten while Proposi-
tion 5.10 says that the restrictions to several disjoints sets Udo not influence each other too
much, even if they are nested. The techniques also allow us to study the dimer model with
Temperleyan boundary conditions, with essentially the same results as in the spanning tree
case.
Before discussing in more details the motivations leading to Theorem 1.1, let us take a
step back and recall briefly some of the history on the UST and dimer model. To the best
of our knowledge, the study of UST goes back to the well known matrix-tree theorem of
Kirchhoff establishing that on any finite graph, number of spanning trees is given by any
cofactor of the Laplacian matrix. This shows that spanning trees are very nice combinatorial
objects, non-trivial but still quite amenable to analysis and with a remarkably simple formula
appearing in the end. Similarly, the earliest result on the dimer model [Mac15] is a formula
for the number a ways to tile an hexagon which is so simple and mysterious the we don’t
resist the temptation to write it :
]{lozenge tiling of hexagon with sides a, b, c}=
a
Y
i=1
b
Y
j=1
c
Y
k=1
i+j+k−1
i+j+k−2.
Strikingly after respectively 160 and 110 years of study, there are still both a great number
of natural unsolved questions on both models and many different and powerful methods
to study them (in part evolved out of the counting arguments above, in part completely
different).
Due to the age and prominence of these models, we will not give a full account of their
history but let us still mention a few results on the aspects of the models that will be used
later. The bijection between the two models was first described by Temperley [Tem74] in
the square lattice and then extended to a more general setting in [KPW00] and then [KS04].
The link between UST and loop-erased random walk (LERW from now on), which serves
2
as the backbone of all the analysis in this paper, was discovered in [Pem91] and refined into
an sampling algorithm in the landmark paper of Wilson [Wil96]. We will build extensively
on [Sch00] which used Wilson’s algorithm to derive many qualitative properties of planar
UST. Of course in this setting we have to mention the convergence of UST to SLE2/SLE8
from [LSW04] (and [YY11] for general planar graphs) which is one of the main motivations
for our statement. Let us also note that in three dimension, LERW is actually one of very
few statistical model where a convergence is proved [Koz07]. On the dimer side, if we don’t
assume anything about the boundary the two main known results are a law of large number
for the height function [CKP01] and a recent local limit [Agg19]. In this paper however
we will only consider so called Temperleyan boundary conditions for which, in a sense, the
limit in the law of large number is just the function 01. In such domain, as mentioned above
Kenyon [Ken00; Ken01] obtained the convergence of fluctuations of the height function in
Z2. This was extended to more general graphs in [BLR19] by exploiting more precisely the
link between dimers and spanning trees.
Let us now come back to Theorem 1.1 and discuss why we would like to prove such a
statement. First let us point out that for the SLE, the effect of changing the domain of
definition on the law of the curve was studied very early (see for example [LSW03]) and
was a key tool to identify the law of the boundary of Brownian motion. Beyond the special
property at κ=8
3and κ= 6, all SLE curves satisfy at least locally Theorem 1.1 and the
Radon-Nycodym derivative is in fact explicit. For the Gaussian free field which describe
the limit of of the dimer height function, since it is (as its name indicated) a Gaussian
process- it is not hard to check with a generalised Girsanov theorem that Theorem 1.1 also
holds, again with an explicit Radon-Nycodym derivative. Together with the scaling limit
results from the previous paragraph, this proves that a version of Theorem 1.1 holds “for
macroscopic quantities” but it natural to ask if microscopic details can still carry some
information. In fact, we believe that specifically for dimers and UST, keeping track of
microscopic information is important because they can be related to several different models
(abelian sandpile, double dimer, xor-Ising, O(1) loops on hexagonal lattice) using bijections
(or measure preserving maps) that rely heavily on these details. In particular for the double
dimer, we plan in a future work to establish Russo-Seymour-Welsh estimates using the
results from this paper.
Finally, since for SLE the result is true for all κ, we believe that analogue of Theorem 1.1
should be true for almost all critical models, and similarly for the extensions of Section 5.
To the best of our knowledge however, even for the critical Ising model the continuity with
respect to boundary condition in Corollary 5.1 is not proved and this is a significant limit
in our understanding because some of our best techniques are analytic and require smooth
boundary conditions.
The rest of the paper is organised as follow. Section 2.1 completes the fully rigorous
statement of Theorem 1.1 by stating the required assumptions on the sequence of graphs
G#δand provides a few properties of the random walk under these assumptions. The rest
of Section 2 contains some background on the objects that will be used later in the paper.
The key known results appearing in the proofs are restated for the sake of completeness
and ease of reference. More precisely we discuss in that order the loop-soup measure, UST
with Wilson’s algorithm and the finiteness theorem, some theory of conformal mappings
with rough boundary conditions and the link between dimers and spanning tree. Sections 3
and 4 contain the proof of the main theorem. Assuming without loss of generality that
D1⊂D2in Theorem 1.1, Section 3 contains the proof of the first part of the statement in
Theorem 1.1 (where in fact only the upper bound is non-trivial), while Section 4 proves the
second statement. Finally Section 5 treats variants of Theorem 1.1 such as the analogous
1On the other hand, because of the arbitrary choices involved in the defintiion of the height function this is
not such a strong condition. In fact for lozenge tiling, [Las21] shows that Temperleyan boundary conditions are
in a sense dense in the set of boundary conditions leading to smooth limit shapes.
3
result for the dimer model or the case where Uis an annulus. Let us emphasize that
the reader should be able to skip all or parts of Section 2 if she is already familiar with
the respective topic. The precise assumptions in Section 2.1 can be ignored if one always
consider G#δ=δZ2which does not make the later arguments significantly easier. Also
Sections 2.4 and 2.5 are only necessary for the extension to the dimer model.
Acknowledgments : AB and BL were supported by ANR DIMERs, grant number
ANR-18-CE40-0033. We benefited from many discussions with colleagues including Misha
Basok, Nathanael Berestycki, C´edric Boutillier, Th´eo Leblanc and Gourab Ray.
2 Assumptions and background
In this section, we state more formally our assumptions with respect to the underlying
graphs we will be studying (Section 2.1), as well as provide a collection of previous results
and definitions needed later in order to make the paper more self contained. A reader
familiar with the respective material can certainly skip the sections from 2.2 to 2.5 and in
fact no important idea from this paper would be lost if the reader skips 2.1 and always take
D#δto be nice subgraphs of δZ2. We finally note that Sections 2.4 and 2.5 are only used
for the extension of Section 5.
2.1 Graphs and random walk
We consider a collection of graphs (G#δ)δ≥0which are oriented, weighted planar graphs
(possibly with self loops and multiples edges). The reader is advised to think of δas a scale
parameter but note that we are not restricting ourself to scaled version of a fixed graph.
Interpreting the weights as transition rates, we obtain a natural continuous time random
walk on G]δ and we denote the law of the walk started from a vertex vby Pv. For an
oriented edge e, we will denote its weight by w(e) and we denote the set of oriented edges
of G#δby Eδ. For ease of notations, we will allow ourself to write an oriented edge from x
to yas x→yas if there was no multiple edges. When using this notation, we will adopt
the convention that w(x→y) = 0 if there is no edge from xto y. Additionally we make the
following assumptions on the collection G#δ.
Figure 1: A schematic representation of the crossing estimate.
Assumption 2.1. •(Good embedding) For all δ, the edges of G#δare embedded in
the plane in such a way that they do not cross each other and are piecewise smooth.
•(Uniformly bounded density) There exists a constant M > 0such that for all δ > 0
and for all z∈C, the number of vertices in the square z+ [0, δ]2is bounded above by
M.
•(Connected) For any δ > 0, the graph G#δis connected in the sense that it is irre-
ducible for the continuous time random walk: for any two vertices uand v, we have
Pu(RW1=v)>0.
4
•(Uniform crossing estimate) We denote by R(resp. R0) the horizontal (resp.
vertical) rectangle [0,3] ×[0,1] (resp. [0,1] ×[0,3]). Let B1=B((1/2,1/2),1/4) be the
starting ball and B2=B((5/2,1/2),1/4) be the target ball. There exist two universal
constants p > 0and δ0>0such that the following is true. For all z∈C, > 0,
v∈B1and δ≤δ0such that z+v∈G#δ, we have
Pv+z(RW hits B2+zbefore exiting R+z)> p. (1)
Additionally, for the results of Section 2.5 and the statements about the dimer model in
Section 5 we will require the following additional assumption
Assumption 2.2. Edges have uniformly bounded winding. More precisely there exists C > 0
such that for any edge e(seen as a curve parameterized with positive speed), for all s, t,
|arg(e0(s)) −arg(e0(t))| ≤ C, where the argument is taken so that this expression is continu-
ous in sand taway from discontinuity of the derivative and with the natural corresponding
convention for jumps.
In the following, it will be useful to have a canonical way to associate a subgraph of G#δ
to a open set, we therefore introduce the following notation. Given Dan open set of the
plane, we let D#δbe the subgraph obtained from the vertices and edges which are in Din
the embedding of G#δ. Note to be precise that the convention is that an edge connecting
two points inside Dbut extincting Dalong the way is not included in D#δ. This is natural
in order to allow slits in D. As is usual we will call connected non-empty open sets domains.
When considering the random walk on D#δ, we will always use wired boundary condi-
tions in the following sense : we add to D#δa “cemetery” point (later called with a slight
abuse of notation ∂D) with an outgoing transition rate of 0. For every edge of G#δwith a
starting point in D#δ, and touching ∂D, we add an edge with the same starting point and
weight going to the cemetery. When considering spanning trees of D#δ, we will always talk
of trees with wired boundary condition and oriented towards the cemetery.
We will also allow ourself to consider a discrete path as the continuous path obtained
by concatenating the edges. With a slight abuse of notation, we will also denote by γ[s, t]
both the subpath of γbetween times sand tand its support. To be consistent with this
convention and our definition of wired boundary condition, we will consider the exit time of
a set Dto be sup{t:γ[0, t]⊂D}, i.e using an edge that starts and ends in Dbut intersects
∂D counts as an exit. To be consistent with the continuous time random walk, we will
sometime write XT=efor an edge eif Tis a time where a jump along the edge eoccurs.
In the following, we will only be interested by (compact) curves up to their time parametri-
sation (in fact note that our assumptions on the random walk are invariant under time
change). When considering the distance between curves, we will always use the L∞distance
up to time reparametrisation, i.e
d(γ1, γ2) = inf
f,g sup
t|γ1(f(t)) −γ2(g(t))|,
where fand gare continuous, strictly increasing, from say [0,1] onto the domain of definition
of γ1and γ2. We will also say that a sequence of rectangles (which are translated and scaled
versions of Rand R0) is -close to a curve γif none of the long sides of the rectangles is
larger than /2, all the starting and target balls match in order and the curve connecting
the center of these balls by straight lines in order is at distance at most /2 of γ.
We now state a few consequences of our assumptions on the walk. First a simple state-
ment saying that the random walk has a positive probability to follow any trajectory.
Lemma 2.3. For any compact curve γfrom [0,1] to C, for any  > 0, there exists p > 0
and a stopping time Tsuch that for all δsmall enough, for all v∈B(γ[0], ), we have
Pv(d(X[0, T ], γ)≤)> p.
5
摘要：

Doesaportionofdimercon gurationdeterminesitsdomainofde nition?AntoineBannierBeno^tLaslierOctober25,2022AbstractCriticalmodelsare,almostbyde nition,supposedtofeaturebothslowdecayofcorre-lationsforlocalobservableswhileretainingsomemixingevenformacroscopicobservables.Astrongversionofthelatterpropertyi...

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