# Does a portion of dimer configuration determines its domain of definition?

Antoine Bannier  Benoît Laslier

###### Abstract

Critical models are, almost by definition, supposed to feature both slow decay
of correlations 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. 1 Introduction
  2. 2 Assumptions and background
    1. 2.1 Graphs and random walk
    2. 2.2 Loop soup
    3. 2.3 Uniform spanning tree
    4. 2.4 Uniformisation and prime ends
    5. 2.5 Dimers, winding, and height function
  3. 3 Upper bound on $\frac{d\nu_{1}}{d\nu_{2}}$
    1. 3.1 Coupling
    2. 3.2 Radon-Nikodym bound
  4. 4 Lower bound on $\frac{d\nu_{1}}{d\nu_{2}}$
    1. 4.1 Mixed Loop-erasing
    2. 4.2 RW in $D_{2}$
    3. 4.3 RW in $D_{1}$
    4. 4.4 loop-erasure
    5. 4.5 Coupling several paths
  5. 5 Extensions and generalisations
    1. 5.1 Continuity with respect to the domains
    2. 5.2 Dimer configurations and height functions
    3. 5.3 Sampling an annulus
    4. 5.4 Continuum limit

##  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 $U,D_{1},D_{2}$ be
three simply connected bounded open sets, with $\overline{U}\subset D_{1}\cap
D_{2}$ and let $U^{\\#\delta},D_{1}^{\\#\delta},D_{2}^{\\#\delta}$ be
discretisations of these sets (ultimately we will want to be as general as
possible here but, for now, it is enough to think that
$U^{\\#\delta}=U\cap\delta\mathbb{Z}^{2}$ and similarly for the others). Our
mixing statement then becomes a testing problem: if one is given the
restriction to $U^{\\#\delta}$ of some UST, can we say whether it was sampled
from $D_{1}^{\\#\delta}$ or $D_{2}^{\\#\delta}$ ? By “fuzzy memory”, we mean
that, even asymptotically as $\delta\to 0$, 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 $\nu_{1},\nu_{2}$ denote the laws of the restrictions to $U$ of wired UST
in $D_{1}^{\\#\delta}$ and $D_{2}^{\\#\delta}$ respectively. For all
$\epsilon>0$, there exits $C=C(U,D_{1},D_{2},\epsilon)>0$ such that for all
$\delta$ small enough

$\nu_{1}(\frac{1}{C}\leq\frac{d\nu_{1}}{d\nu_{2}}\leq C)\geq
1-\epsilon,\qquad\nu_{2}(\frac{1}{C}\leq\frac{d\nu_{1}}{d\nu_{2}}\leq C)\geq
1-\epsilon.$

Essentially one can think of that result as stating that $\nu_{1}$ and
$\nu_{2}$ are mutually absolutely continuous in a way which is independent of
$\delta$.

In fact Theorem 1.1 and the techniques used to prove it will provide a number
of extensions 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 Proposition 5.10 says that the
restrictions to several disjoints sets $U$ do 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 :

$\sharp\\{\text{lozenge tiling of hexagon with sides
$a,b,c$}\\}=\prod_{i=1}^{a}\prod_{j=1}^{b}\prod_{k=1}^{c}\frac{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 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 $0$111On 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.. In such domain, as mentioned above Kenyon
[Ken00, Ken01] obtained the convergence of fluctuations of the height function
in $\mathbb{Z}^{2}$. 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
$\kappa=\frac{8}{3}$ and $\kappa=6$, 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 $\kappa$, 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 $G^{\\#\delta}$ and 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 $D_{1}\subset D_{2}$ in 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 result for the dimer model or the case where $U$ is 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
$G^{\\#\delta}=\delta\mathbb{Z}^{2}$ which 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, Cédric Boutillier, Théo 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
$D^{\\#\delta}$ to be nice subgraphs of $\delta\mathbb{Z}^{2}$. 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 $(G^{\\#\delta})_{\delta\geq 0}$ which are
oriented, weighted planar graphs (possibly with self loops and multiples
edges). The reader is advised to think of $\delta$ as 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 $G^{\sharp\delta}$ and we denote the law of the walk
started from a vertex $v$ by $\mathbb{P}_{v}$. For an oriented edge $e$, we
will denote its weight by $w(e)$ and we denote the set of oriented edges of
$G^{\\#\delta}$ by $E_{\delta}$. For ease of notations, we will allow ourself
to write an oriented edge from $x$ to $y$ as $x\to y$ as if there was no
multiple edges. When using this notation, we will adopt the convention that
$w(x\to y)=0$ if there is no edge from $x$ to $y$. Additionally we make the
following assumptions on the collection $G^{\\#\delta}$.

Figure 1: A schematic representation of the crossing estimate.

######  Assumption 2.1.

  * •

(Good embedding) For all $\delta$, the edges of $G^{\\#\delta}$ are 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 $M>0$ such that for all
$\delta>0$ and for all $z\in\mathbb{C}$, the number of vertices in the square
$z+[0,\delta]^{2}$ is bounded above by $M$.

  * •

(Connected) For any $\delta>0$, the graph $G^{\\#\delta}$ is connected in the
sense that it is irreducible for the continuous time random walk: for any two
vertices $u$ and $v$, we have $\mathbb{P}_{u}(RW_{1}=v)>0$.

  * •

(Uniform crossing estimate) We denote by $\mathcal{R}$ (resp.
$\mathcal{R}^{\prime}$) the horizontal (resp. vertical) rectangle
$[0,3]\times[0,1]$ (resp. $[0,1]\times[0,3]$). Let $B_{1}=B((1/2,1/2),1/4)$ be
the starting ball and $B_{2}=B((5/2,1/2),1/4)$ be the target ball. There exist
two universal constants $p>0$ and $\delta_{0}>0$ such that the following is
true. For all $z\in\mathbb{C}$, $\epsilon>0$, $v\in\epsilon B_{1}$ and
$\delta\leq\epsilon\delta_{0}$ such that $z+v\in G^{\\#\delta}$, we have

$\mathbb{P}_{v+z}(RW\mbox{ hits }\epsilon B_{2}+z\mbox{ before exiting
}\epsilon\mathcal{R}+z)>p.$ (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 $s,t$, $|\arg(e^{\prime}(s))-\arg(e^{\prime}(t))|\leq C$, where the
argument is taken so that this expression is continuous in $s$ and $t$ away
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 $G^{\\#\delta}$ to a open set, we therefore introduce the
following notation. Given $D$ an open set of the plane, we let $D^{\\#\delta}$
be the subgraph obtained from the vertices and edges which are in $D$ in the
embedding of $G^{\\#\delta}$. Note to be precise that the convention is that
an edge connecting two points inside $D$ but extincting $D$ along the way is
_not_ included in $D^{\\#\delta}$. 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 $D^{\\#\delta}$, we will always use wired
boundary conditions in the following sense : we add to $D^{\\#\delta}$ a
“cemetery” point (later called with a slight abuse of notation $\partial D$)
with an outgoing transition rate of $0$. For every edge of $G^{\\#\delta}$
with a starting point in $D^{\\#\delta}$, and touching $\partial D$, we add an
edge with the same starting point and weight going to the cemetery. When
considering spanning trees of $D^{\\#\delta}$, 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 $\gamma[s,t]$ both the subpath of $\gamma$ between times $s$
and $t$ and its support. To be consistent with this convention and our
definition of wired boundary condition, we will consider the exit time of a
set $D$ to be $\sup\\{t:\gamma[0,t]\subset D\\}$, i.e using an edge that
starts and ends in $D$ but intersects $\partial D$ counts as an exit. To be
consistent with the continuous time random walk, we will sometime write
$X_{T}=e$ for an edge $e$ if $T$ is a time where a jump along the edge $e$
occurs.

In the following, we will only be interested by (compact) curves up to their
time parametrisation (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 $L^{\infty}$ distance up to time reparametrisation, i.e

$d(\gamma_{1},\gamma_{2})=\inf_{f,g}\sup_{t}|\gamma_{1}(f(t))-\gamma_{2}(g(t))|,$

where $f$ and $g$ are continuous, strictly increasing, from say $[0,1]$ onto
the domain of definition of $\gamma_{1}$ and $\gamma_{2}$. We will also say
that a sequence of rectangles (which are translated and scaled versions of
$\mathcal{R}$ and $\mathcal{R}^{\prime}$) is $\epsilon$-close to a curve
$\gamma$ if none of the long sides of the rectangles is larger than
$\epsilon/2$, 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 $\epsilon/2$ of $\gamma$.

We now state a few consequences of our assumptions on the walk. First a simple
statement saying that the random walk has a positive probability to follow any
trajectory.

######  Lemma 2.3.

For any compact curve $\gamma$ from $[0,1]$ to $\mathbb{C}$, for any
$\epsilon>0$, there exists $p>0$ and a stopping time $T$ such that for all
$\delta$ small enough, for all $v\in B(\gamma[0],\epsilon)$, we have

$\mathbb{P}_{v}(d(X[0,T],\gamma)\leq\epsilon)>p.$

###### Proof.

Note that with the uniform topology on curves up to re-parametrisation, any
$\gamma$ curve can be approximated by a finite concatenation of straight lines
up to precision $\epsilon$. Indeed define a sequence of times
$T_{i}=\inf\\{t>T_{i-1}:\gamma(t)\not\in B(\gamma(T_{i-1},\epsilon/2))\\}$. By
uniform continuity of $\gamma$, this sequence must contain finitely many terms
and by construction the distance between $\gamma$ and the piecewise linear
curve connecting the $\gamma(T_{i})$ in order is at most $\epsilon$.

Clearly any piecewise linear curve can be approximated by a finite sequence of
rectangles and, by uniform crossing and the Markov property, the random walk
has a positive probability to cross all of them in order. This concludes the
proof. ∎

By considering the possibility that a walk does a “full turn” in an annulus,
we deduce a Beurling estimate (Lemma 4.17 in [BLR19]).

######  Lemma 2.4 (Non quantitative Beurling estimate).

There exists $\alpha>0$ such that for all $r,R$ with $R\geq 2r$, for all
$\delta$ small enough, for all $v\in G^{\\#\delta}$ and for any set $A$
connecting $\partial B(v,r)$ to $\partial B(v,R)$,

$\mathbb{P}_{v}(RW\text{ exits }B(v,R)\text{ before hitting
}A)<(\frac{r}{R})^{\alpha}.$

We will also need a dual statement about the probability that the random walk
first exits a domain from a marked point.

######  Lemma 2.5 (Harmonic measure bound).

There exists $\alpha$ such that for all $r,R$ with $R\geq 2r$, for all
$\delta$ small enough, for all simply connected open set $D$, and all points
$v,w$ with $v\in D^{\\#\delta}$ and $w\in\partial D$ and $\lvert v-w\rvert>R$,
we have

$\mathbb{P}_{v}(\text{ RW exits $D$ inside $B(w,r)$})<(\frac{r}{R})^{\alpha}.$

The proof of this estimate is identical to the Beurling estimate with just $w$
taking the role of infinity so we leave details to the reader.

Arguments similar to the Beurling estimate prove that discrete harmonic
functions must satisfy the Harnack inequality. Together with the
representation of a walk conditioned on its endpoint as an h-transform, this
implies that the uniform crossing also holds for walks conditioned on their
exit point from a domain, at least as long as the rectangle that must be
crossed stays far from the boundary (see Lemma 4.4 in [BLR19] which is stated
for annuli but only uses the Harnack inequality).

######  Lemma 2.6 (Conditional uniform crossing).

Let $\mathcal{R}$, $B_{1}$, $B_{2}$ be as in the uniform crossing assumption.
For all $\alpha>0$ there exists $p>0$ and $\delta_{0}>0$ such that the
following holds. For all $z\in\mathbb{C}$, $\epsilon>0$,
$\delta\leq\epsilon\delta_{0}$, $v\in(\epsilon B_{1}+z)^{\\#\delta}$, $D$ such
that $d(\partial\epsilon\mathcal{R}+z,\partial D)>\alpha\epsilon$ and en edge
$w$ such that $\mathbb{P}_{v}(X(T_{D})=w)>0$,

$\mathbb{P}_{v+z}(X\mbox{ hits }\epsilon B_{2}+z\mbox{ before exiting
}\epsilon\mathcal{R}+z|X(T_{D})=w)>p,$

where $T_{D}$ is the exit time from $D$.

Finally we will need an estimate saying that if a random walk is conditioned
to exit at a certain point and starts close to it, then it is unlikely to ever
go far away. If $D$ is an open set, denote by $T_{D}$ the exit time from that
set (in the sense mentioned in the beginning of the section).

######  Lemma 2.7.

There exists $C$ and $p>0$ such that for any open set $D$ with a smooth
boundary, for all $r>0$ small enough, and all $\delta$ small enough the
following holds. For all $v$ with $v\in D^{\\#\delta}$ and all edges $w$ such
that $\mathbb{P}(X_{T_{D}}=w)>0$ and $2r\geq d(v,\partial D)\geq r$, we have

$\mathbb{P}_{v}(X[0,T_{D}]\subset B(w,Cr)|X(T_{D})=w)>p.$

Furthermore $p$ can be chosen close to $1$ by taking $C$ large enough.

###### Proof.

Fix $D,v,w$ and $r$ as in the statement of the lemma, where the condition on
the size of $r$ will be made precise later. Let us also write
$h(u):=\mathbb{P}_{u}(X(T_{D})=w)$ and note that this is harmonic for the
random walk on $D^{\\#\delta}$. By the Markov property and Beurling’s estimate

$\displaystyle\mathbb{P}_{v}(X[0,T_{D}]\not\subset$ $\displaystyle
B(w,Cr),X(T_{D})=w)$ $\displaystyle\leq\mathbb{P}_{v}(X[0,T_{D}]\not\subset
B(w,Cr)).\max_{\partial B(x,Cr)^{\\#\delta}\cap D^{\\#\delta}}h(u)$
$\displaystyle\leq\frac{1}{(C-1)^{\alpha}}\max_{\partial
B(x,Cr)^{\\#\delta}\cap D^{\\#\delta}}h(u),$

where we use a maximum since $\partial B(x,Cr)^{\\#\delta}\cap D^{\\#\delta}$
is finite. Let $\gamma$ be a path in $B(w,Cr)\cap D$ connecting a point where
this maximum is reached to $w$ and along which $h$ is increasing. Note that by
harmonicity of $h$,

$\mathbb{P}_{v}(X[0,T_{D}]\not\subset
B(w,Cr),X(T_{D})=w)\leq\frac{1}{(C-1)^{\alpha}}\frac{h(v)}{\mathbb{P}_{v}(X_{T_{D}\wedge
T_{\gamma}}\in\gamma)},$

where $T_{\gamma}$ is the hitting time of $\Gamma$. Note that $\gamma$
separates $B(w,Cr)\cap D$ into exactly two connected components, call $A_{1}$
the component containing $u$ and $A_{2}$ the other one. Since $D$ is smooth,
if $r$ is chosen small enough $\partial D$ is almost linear in a neighbourhood
of $w$ and $\partial A_{2}\cap\partial D$ contains an arc connecting $w$ and a
point on $\partial B(w,Cr)$. By Lemma 2.3, the random walk starting in $v$ has
a positive probability $p>0$ (independent of $C$) to follow $\partial
B(w,|v-w|)$ in the direction leading to $\partial A_{2}$ up to an error $r/2$
and on that event we must have $T_{\gamma}<T_{D}$. Overall,

$\mathbb{P}_{v}(X[0,T_{D}]\not\subset
B(w,Cr),X(T_{D})=w)\leq\frac{1}{(C-1)^{\alpha}}\frac{h(v)}{p},$

which concludes. ∎

###  2.2 Loop soup

In this section, we recall the definition of the random walk loop soup measure
as well as its relation to the loop-erased random walk. We also provide a
basic estimate on the mass of macroscopic loops in the measure. We will keep
the exposition short and refer to chapter 9 in [LL10] for much more details on
this topic.

Let us denote by $q(v\to v^{\prime})$ the discrete transition rate from $v$ to
$v^{\prime}$, i.e the edge weights normalised such that
$\sum_{v^{\prime}}q(v\to v^{\prime})=1$. We call a finite sequence
$\gamma=(v_{0},\ldots,v_{n})$ such that $v_{0}=v_{n}$ a _rooted loop_ and call
$v_{0}$ its root. We call length of the loop simply its number of steps, i.e
$\lvert(v_{0},\ldots,v_{n})\rvert=n$. Note that there is no restriction to
simple loops and that the length of a loop is can be much smaller than the
size of its support if one has many multiple point. An _unrooted loop_ (which
we also just call a loop later) is an equivalence class of rooted loop by
cyclical permutation.

######  Definition 2.8.

The _rooted loop measure_ $\Lambda=\Lambda^{\\#\delta}$ on $G^{\\#\delta}$ is
the measure on rooted loops defined as

$\Lambda(v_{0},\ldots,v_{n})=\frac{\prod_{i=0}^{n-1}q(v_{i}\to v_{i+1})}{n}.$

The unrooted loop measure is the image of the rooted measure by the natural
“forgetting the root” map222Typically the mass of an unrooted loop is the
product of transition probabilities but there are small complications if a
loop goes several times around the same path.. We call _loop soup_ of
$G^{\\#\delta}$ the Poisson process with intensity given by the unrooted loop
measure.

The loop measure on $D^{\\#\delta}$ for a subgraph $D^{\\#\delta}$ is simply
the restriction of $\Lambda$ to loops that do not exit $D^{\\#\delta}$ and the
loop soup on $D^{\\#\delta}$ is similarly the restriction of the loop soup.

In this paper, the loop soup will be used to describe the conditional law of a
random walk given its loop-erasure.

######  Construction 2.9.

Fix a subgraph $D^{\\#\delta}$, a path $\gamma=(\gamma_{1},\ldots,\gamma_{n})$
from some $v\in D^{\\#\delta}$ to $\partial D^{\\#\delta}$ and a finite
collection of loops $(\ell_{k})_{k\geq 0}$. We will define a path $X$ as
follow.

  * •

Apply a uniform permutation of the indices to the finite sequence $(\ell_{k})$
(we however keep the same notation for simplicity).

  * •

Define $K_{1}=\\{k:\gamma_{1}\in\ell_{k}\\}$, for each $k\in K_{1}$ let
$\tilde{\ell}_{k}$ denote a version of $\ell_{k}$ rooted at $\gamma_{1}$
(choose uniformly over all possibilities if necessary) and define
$L_{1}=\bigoplus_{k\in K_{1}}\tilde{\ell}_{k}$, i.e $L_{1}$ is the
concatenation of all the loops that intersect $\gamma_{1}$. Note that when
concatenating rooted loops, one must omit some repetition of the root but we
hope the definition is clear.

  * •

Define $K_{2}=\\{k:k\not\in K_{1},\gamma_{2}\in\ell_{k}\\}$ and set
$L_{2}=\bigoplus_{k\in K_{2}}\tilde{\ell}_{k}$ with similar notation as above.

  * •

Define $K_{3}=\\{k:k\not\in K_{1}\cup K_{2},\gamma_{3}\in\ell_{k}\\}$ and so
on up to $K_{n_{1}}$ and $L_{n-1}$.

  * •

Finally we set $X=\big{(}\bigoplus_{i=1}^{n-1}L_{i}\big{)}\oplus\gamma_{n}$.
Note that the edges of $\gamma$ appear as the transitions from $L_{i}$ to
$L_{i+1}$.

The above procedure is motivated by the following result which can be found
for example in [LJ08] as Proposition 28 together with Remark 21.

######  Lemma 2.10.

Fix $D^{\\#\delta}$ together with a path $\gamma$ ending in $\partial
D^{\\#\delta}$ and apply the construction 2.9 to a sample $(\ell_{k})$ of the
loop soup on $D^{\\#\delta}$. The law of the resulting $X$ is given by the
conditional law of a random walk stopped when it exits $D^{\\#\delta}$ given
that it’s loop-erasure if $\gamma$.

######  Remark 2.11.

There is a converse statement which says that if one sample a wired uniform
spanning tree of $D^{\\#\delta}$ using Wilson’s algorithm (see next section),
then the set of erased loops forms in a sense an instance of the loop soup of
$D^{\\#\delta}$. There are however a few subtle points in such a statement,
mainly in how to go from the loops $L_{i}$ to the $\ell_{k}$ in the previous
notations. We do not go into more details here since we will not need the
converse statement.

The last ingredient needed is a bound on the mass of large loops. For general
planar graphs satisfying uniform crossing, this was obtained in Corollary 4.22
from [BLR22].

######  Lemma 2.12.

For all $\epsilon>0$, there exits $C>0$ and $\delta_{0}$ such that for all
$r>0$, for all $\delta\leq r\delta_{0}$, for all $z\in\mathbb{C}$ the
following holds. Let $\mathcal{L}(\epsilon,r,z,\delta)$ be the set of loops
staying in $B(z,r)^{\\#\delta}$ and with diameter greater that $\epsilon r$,
we have

$\Lambda(\mathcal{L}(\epsilon,r,z,\delta))\leq C.$

###  2.3 Uniform spanning tree

We now turn to the main object from this paper, and while we assume that the
reader should be familiar with it, we recall briefly the main properties that
will be needed in the rest of the paper.

In an oriented setting as ours, a spanning tree of a finite graph $G$ rooted
at some vertex $\partial$ is a subset of oriented edges $\mathcal{T}$ which
contains no cycle and such that, in $\mathcal{T}$, $\partial$ has no outgoing
edge and every other vertex has a single outgoing edge in $\mathcal{T}$. It is
easy to see that from any vertex $v\neq\partial$, following the outgoing edges
in order defines a simple path from $v$ to $\partial$ which we naturally call
the branch of $\mathcal{T}$ starting in $v$. This branch will in general be
denoted $\gamma_{v}$. In this paper we will always consider graphs of the form
$D^{\\#\delta}$ with wired boundary condition and root the tree at the wired
vertex. The uniform spanning tree (UST) is the measure such that

$\mathbb{P}(\mathcal{T})\propto\prod_{(vv^{\prime})\in\mathcal{T}}q(v\to
v^{\prime}).$

######  Remark 2.13.

In the most classical case of an un-oriented graph (or equivalently a
reversible Markov chain), forgetting the orientation gives a law on un-
oriented trees which is independent of the choice of root. This does not hold
with general oriented graphs however.

It is well known that the UST can be sampled through Wilson’s algorithm which
we recall rapidly for completeness. Start with an initial trivial tree
$\mathcal{T}_{0}=\\{\partial\\}$ and proceed by induction, constructing
subtrees $\mathcal{T}_{i}$ as follows. Given $\mathcal{T}_{i}$, pick any
vertex (not in $\mathcal{T}_{i}$ otherwise the step is trivial) and starts a
random walk from it until it hits a vertex of $\mathcal{T}_{i}$. Erase the
loops of this walk (say in forward time) to obtain a simple path and set
$\mathcal{T}_{i+1}$ as the union of this path and $\mathcal{T}_{i}$.

One of our key tools to understand this sampling procedure and the geometry of
the UST in general is Schramm’s finiteness theorem [Sch00] which intuitively
says that after a large but finite (and independent of the underlying mesh
size) number of steps in the algorithm, all the macroscopic structure of the
tree is already determined and only small local details still need to be
sampled. Here we use the following version of that theorem which goes back to
[BLR19], Lemma 4.18.

######  Lemma 2.14.

Fix $D$ a simply connected bounded domain and $U$ open such that
$\bar{U}\subset D$. For all $\epsilon>0$, there exits $k\in\mathbb{N}$ such
that for all $\delta$ small enough, one can find $k$ vertices of
$D^{\\#\delta}$ within distance $\epsilon$ of $U$ such that the following
holds. Suppose that we start by sampling the branches from $v_{1}$ up to
$v_{k}$ in Wilson’s algorithm and let $\mathcal{T}_{k}$ be the corresponding
subtree. There is a way to choose points $v_{k+1},\ldots$ where we still only
consider points in within distance $\epsilon$ of $U$ and eventually exhaust
all the vertices of $U$ (call the resulting subtree $\mathcal{T}_{U}$) so
that, except on a event of probability at most $\epsilon$,

  * •

No connected component of $\mathcal{T}_{U}\setminus\mathcal{T}_{k}$ has
diameter more than $\epsilon$,

  * •

for all $i\geq k+1$, the random walk started from $v_{i}$ hits
$\mathcal{T}_{i-1}$ before exiting $B(v_{i},\epsilon)$.

Essentially, this statement means that in order to sample all the branches
starting in $U$ of an UST, it is enough to understand well the behaviour of
the first $k$ branches (which unfortunately have to start in a neighbourhood
of $U$) and everything else will be made of very small subtrees attached to
these branches first few branches. Furthermore and crucially for our purpose,
these small subtrees are generated by erasing walks that never went far from
$U$ and are therefore likely easy to couple between different domains.

###  2.4 Uniformisation and prime ends

In the future, we want our results to be usable out of the box say for the law
of a spanning tree conditioned on one of its branch. By Wilson’s algorithm,
this just means that we want to be able to apply our results on domains
slitted by a rough curve, including discussing winding of curves in such
domain for the application to dimers. For that purpose, we now introduce a few
elements on uniformisation of domains with rough boundaries. Unless specified
otherwise, all results are found in [Pom92] chapter 2 and they will only be
used in Section 5.

######  Definition 2.15.

A closed set $A\subset\mathbb{C}$ is locally connected if for all $\epsilon$,
there exists $\eta$ such that for all $x,y\in A$, if $|x-y|\leq\eta$ then
there exists a connected subset $B$ of $A$ such that $x,y\in B$ and
$\operatorname{diam}(B)\leq\epsilon$.

######  Definition 2.16.

Let $D$ be a domain, the diameter distance $d_{D}$ on $D$ is defined as

$d_{D}(x,y)=\inf_{\text{$\gamma$ connecting $x$ to
$y$}}\operatorname{diam}(\gamma),$

where in the above infimum $\gamma$ must be a continuous path in $D$. the
function $d_{D}$ is a distance on $D$ and we let $\overline{D}_{d}$ denote the
completion of $D$ for this distance and
$\partial_{d}D=\overline{D}_{d}\setminus D$.

As an aside, note that one could also define a natural intrinsic distance
using the length of curves instead of their diameter. A moment of though shows
however that the length distance can be much finer than the diameter one close
to a rough boundary and for our purpose we really need to consider the
diameter.

######  Theorem 2.17.

Let $D$ be a bounded simply connected domain, and let $f$ denote any conformal
map mapping $\mathbb{D}$ to $D$. The following five conditions are equivalent

  * •

$f$ has a continuous extension to $\overline{\mathbb{D}}$,

  * •

$f$ extends to an homeomorphism from $\overline{\mathbb{D}}$ to
$\overline{D}_{d}$.

  * •

$\partial D$ can be parametrised as a continuous loop, i.e there is a
continuous $\gamma$ from the unit circle to $\mathbb{C}$ such that $\partial
G=\gamma(\mathbb{S}_{1})$,

  * •

$\partial D$ is locally connected,

  * •

$\mathbb{C}\setminus D$ is locally connected.

In particular we can always assume in the third line that the parametrisation
of the boundary is of the form $\gamma(e^{i\theta})=f(e^{i\theta})$ where $f$
is conformal. The second statement in this theorem is not given in [Pom92] but
can be found in [Her12].

In this context, we will call an element of $\partial_{d}D$ a prime end. This
is a slight abuse of terminology since the true notion of prime end is more
general, we refer the interested reader to [Pom92] for the actual definition
and the fact that it matches our usage in the locally connected case.

Clearly the diameter distance on $D$ is finer that the restriction to $D$ of
the usual euclidean distance and therefore there is a canonical onto map from
$\partial_{d}D$ to $\partial D$, i.e each prime end is associated to a single
boundary point but several prime end can share the same image (for examples
the two sides of a slit are different prime ends). It is also easy to see
using uniform continuity that if $\gamma$ is a continuous path from $[0,1]$ to
$\mathbb{C}$ such that $\gamma[0,1)\subset D$ then $\gamma$ can be extended to
a continuous path from $[0,1]$ to $\overline{D}_{d}$. In particular any such
curve ending on $\partial D$ determines a unique prime end which will be
important later in the paper.

Finally, let us introduce a notion of distance between domains with a marked
prime end that will be useful in Section 5.

######  Definition 2.18.

Let $A$, $B$ be two bounded simply connected domains with locally connected
boundaries and let $x\in\partial_{d}A$ and $x\in\partial_{d}B$ be two prime
ends. For $\epsilon>0$, we say that the marked domains $(A,x)$ and $(B,y)$ are
$\epsilon$-close if there exists an homomorphism
$f:\overline{A}_{d}\to\overline{B}_{d}$ such that $f(x)=y$, for all $a\in A$,
$|f(a)-a|\leq\epsilon$ and for all $b\in B,|f^{-1}(b)-b|\leq\epsilon$.

Let us note that we will not actually need to show that the above definition
defines a distance. Thanks to Theorem 2.17, we have a simple (and trivial)
criterion in terms on uniformisation maps for two marked domains to be close.

######  Lemma 2.19.

Let $A,B,x,y$ be as above and let $f:\mathbb{D}\to A$ and $g:\mathbb{D}\to B$
be two uniformisation maps such that $f(1)=x$ and $g(1)=y$. If $\lVert
f-g\rVert_{L^{\infty}(\overline{\mathbb{D}})}\leq\epsilon$ then $(A,x)$ and
$(B,y)$ are $\epsilon$-close.

As shown in Figure 2, the notion of distance from Definition 2.18 cannot be
simply rewritten in terms of Caratheodory’s topology or in terms of the
distance between the boundary seen as curve so we unfortunately cannot provide
a simple direct geometric description of it. However we can still obtain the
following non-uniform continuity statement

Figure 2: Two slitted versions of the unit disc with a marked prime end
denoted by an arrow. Note that both domains are close to the regular unit disc
for Caratheodory’s topology with respect to the center of the disc. Also, when
seen as curve parametrised from the marked prime end, the boundary of these
domains are close. Yet the two marked domains are _not_ close in the sense of
Definition 2.18

######  Lemma 2.20.

Fix $A$ a bounded simply connected domain with locally connected boundary and
$x\in\partial_{d}A$ a prime end. For every $\epsilon>0$, there exists $\eta>0$
such that for all bounded simply connected domain with locally connected
boundary $B$ and prime end $y\in\partial_{d}B$ the following holds. Let
$\gamma_{A}$ and $\gamma_{B}$ be two parametrisation of $\partial A$ and
$\partial B$ as close curves starting from $x$ and $y$ respectively. If the
uniform distance up to reparametrisation between $\gamma_{A}$ and $\gamma_{B}$
is smaller than $\eta$ then $(A,x)$ and $B(y)$ are $\epsilon$-close.

In other word, every pair $(A,x)$ is a continuity point for the map between
the distance on boundary curves to the one from Definition 2.18. Let us also
note that the arbitrary nature of the choice of parametrisation for
$\gamma_{A}$ and $\gamma_{B}$ above is irrelevant with our choice of distance.

###### Proof.

Fix $A,B,x,y,\gamma_{A},\gamma_{B}$ as above together with some $z\in A$.
Since as noted above the choice of $\gamma_{A}$ is irrelevant, we can assume
without loss of generality that $\gamma_{A}:[0,2\pi]\to\partial A$ is obtained
from the map conformal map $f:\mathbb{D}\to A$ sending $0$ to $z$ and $1$ to
$x$ and that $\gamma_{B}$ is also defined on $[0,2\pi]$ with
$\lVert\gamma_{A}-\gamma_{B}\rVert_{\infty}\leq\eta$ for some $\eta$ to be
taken small enough later. In fact let us already assume that
$\eta<d(z,\partial A)$ and note that this imply that $z\in B$. Finally let
$\operatorname{Harm}_{A}(\gamma_{A}[0,s])$ denote the harmonic measure in $A$
seen from $z$ of $\gamma_{A}[0,s]$ seen as a set of prime end, with a similar
notation for $B$.

Consider a planar Brownian motion $B_{t}$ started from $z$ and let $\tau_{A}$
(resp. $\tau_{B}$) be its exit time from $A$ (resp. $B$). Also let
$T=\inf\\{t>0:d(B_{t},\partial A)\leq 2\eta\\}$. Note that by Beurling’s
estimate (Lemma 2.4) there exits $\alpha>0$ (independent of $A$, $B$) such
that $\mathbb{P}(d_{A}(B_{T}-B_{\tau_{A}})\geq\sqrt{\eta})\leq\eta^{\alpha}$
and similarly for $B$. In particular for all $s>0$, we have

$\mathbb{P}(d_{A}(X_{T},\gamma_{A}[0,s])\geq\sqrt{\eta})\leq\mathbb{P}(X_{\tau}\not\in\gamma_{A}[0,s])+\eta^{\alpha}\leq
1-s+\eta^{\alpha}.$

On the other hand, by Theorem 2.17 $\gamma^{-1}_{A}$ seen as a curve from
$\overline{A}_{d}$ to $\overline{B}$ is continuous but therefore also
uniformly continuous. In particular we can find $\beta>0$ (depending only on
$A$ and going to $0$ with $\eta$) such that for all $s,s^{\prime}$ with
$|s-s^{\prime}|\geq\beta$, we have
$d_{A}(\gamma_{A}(s),\gamma_{A}(s^{\prime}))\geq 2\sqrt{\eta}$. Again using
Beurling’s estimate, we obtain that for all $s\geq 2\beta$

$\mathbb{P}(d_{A}(X_{T},\gamma_{A}[\beta,s-\beta])\leq\sqrt{\eta})\leq\mathbb{P}(X_{\tau}\in\gamma_{A}[0,s])+\eta^{\alpha}\leq
s-\eta^{\alpha}.$

Combining the two above estimates, we see that for all $s\geq 2\beta$,

$s-\eta^{\alpha}\leq\mathbb{P}(d_{A}(X_{T},\gamma_{A}[0,s])\leq\sqrt{\eta})\leq
s+2\beta+\eta^{\alpha}.$

Let us now turn to estimates related to $B$. Suppose that
$d_{A}(X_{T},\gamma_{A}[\beta,s-\beta]\leq\sqrt{\eta}$, then by definition of
$\beta$, $d_{eucl}(X_{T},\partial A\setminus\gamma_{A}[0,s])\geq\sqrt{\eta}$
but then we also have $d_{eucl}(X_{T},\partial
B\setminus\gamma_{B}[0,s])\geq\sqrt{\eta}-\eta$ and by Beurling

$\mathbb{P}(X_{\tau_{B}}\in\gamma_{B}[0,s]|d_{A}(X_{T},\gamma_{A}[\beta,s-\beta]\leq\sqrt{\eta})\geq
1-2\eta^{\alpha},$

and therefore for all $s$,

$\operatorname{Harm}_{B}(\gamma_{B}[0,s])\geq(1-2\eta^{\alpha})(s-2\beta-\eta^{\alpha}).$

Since the bounds are arbitrary, we have a corresponding upper bound by
considering the complementary event.

To conclude we will apply Lemma 2.19. To that end, let $\tilde{\gamma}_{B}$ be
the reparametrisation of $\gamma_{B}$ by harmonic measure, i.e such that
$\operatorname{Harm}_{B}(\tilde{\gamma}_{B}[0,s])=s$ for all $s$. Note that
this implies that $\tilde{\gamma}_{B}$ is the boundary extension of the
conformal map $g:\mathbb{D}\to B$ sending $0$ to $z$ and $1$ to $y$. On the
other hand, combining the uniform continuity of $\gamma_{A}$ and the previous
bound, it is easy to see that we can find $\epsilon$ independent of $B$ and
going to $0$ with $\eta$ such that
$\lVert\gamma_{A}-\tilde{\gamma}_{B}\rVert\leq\epsilon$. By the maximum
principle, this shows that $|f-g|\leq\epsilon$ on $\overline{\mathbb{D}}$
which concludes by Lemma 2.19. ∎

###  2.5 Dimers, winding, and height function

In this section we recall some simple facts about winding of open curves and
then present the connection between uniform spanning tree and dimer model. We
use the terminology from [BLR22].

######  Definition 2.21.

Let $\gamma:[0,1]\to\mathbb{C}$ be a continuous curve. For
$z\not\in\gamma([0,1])$, we define the (topological) winding of $\gamma$
around $z$ to be

$W(\gamma,z)=\arg(\gamma(1)-z)-\arg(\gamma(0)-z),$

where the argument is taken so that it is continuous along $\gamma$. If
$\gamma$ admits a tangent at its starting, we set
$W(\gamma,\gamma(0))=\arg(\gamma(1)-\gamma(0))-\lim_{\epsilon\to
0}\arg(\gamma(\epsilon)-\gamma(0))$ and similarly for the ending point.

For smooth curve, there is also a natural “‘intrinsic” definition of winding
that does not need a reference point since one can track the argument of the
derivative. It is related to our definition though the following proposition:

######  Proposition 2.22.

Let $\gamma:[0,1]\to\mathbb{C}$ be a smooth injective curve such that
$\gamma^{\prime}(t)\neq 0$ for all $t\in[0,1]$. We have

$W(\gamma,\gamma(0))+W(\gamma,\gamma(1))=\arg(\gamma^{\prime}(1))-\arg(\gamma^{\prime}(0)),$

where the arguments on the right hand side are continuous along
$\gamma^{\prime}$.

Thanks to this proposition, we can generalise the intrinsic notion of winding
to any simple curve which is regular close to its endpoints even if it is
globally very rough. Also since $W(\gamma,z)$ is a topological quantity, it is
not hard to see that the formula also holds for limits of simple curves, i.e
we can think of $W(\gamma,\gamma(0))+W(\gamma,\gamma(1))$ as an intrinsic
winding for any non self-crossing curve.

Let us now describe the first map from a spanning tree to a dimer
configuration in the context of the discretization procedure of Section 2.4,
see [KPW00] for the original description of that map and [BLR22] for a
detailed analysis of the height function in terms of winding.

We start with a simply connected bounded open set $D$. Fix a _reference prime
end_ of $D$, associated to a point $x_{{\text{ref}}}$ of $\partial D$ and a
simple path $\gamma_{{\text{ref}}}$ starting from $x_{{\text{ref}}}$ and going
to infinity. For simplicity, let us assume that $\gamma_{{\text{ref}}}$ only
intersects $\overline{D}$ at $x_{{\text{ref}}}$ (in particular
$x_{{\text{ref}}}$ this imply that $x_{{\text{ref}}}$ was not chosen on a
slit). The interested reader can check that we could drop this assumption by
interpreting $\gamma_{{\text{ref}}}$ correctly as a path intersecting but not
crossing $\partial D$. Also let us assume that $\gamma_{{\text{ref}}}$ merges
with the positive real axis at some point and goes to infinity along the
positive real axis. It will be clear from the construction later that we do
not lose generality assuming this and it will be convenient when comparing
different domains later.

Recall that we see $D^{\\#\delta}$ as a wired graph where every edge of
$G^{\\#\delta}$ intersecting $\partial D$ was replaced by an edge directed to
a single boundary vertex. We now give a slightly different interpretation of
the same graph as follows. For each edge (which remember is always oriented)
intersecting $\partial D$, we look at the path from the starting point up to
the first intersection with $\partial D$. As noted after Theorem 2.17 we can
see these paths as ending at prime ends so we add a single vertex for each of
those prime end.

Since $\partial D$ is locally connected, we can see it as a continuous curve
starting and ending at the reference prime end. This produces an ordering of
all the extra vertices added in the previous step starting just after the
marked prime end and we add an oriented edge from each prime end to the next
one, following $\partial D$, adding if necessary an extra vertex at the marked
prime end. By design, this construction is equivalent in terms of random walk
to the original one : we have simply replaced the single boundary point by a
path following the boundary with only deterministic transitions until it
reaches $x_{\text{ref}}$.

Now consider the planar dual of $D^{\\#\delta}$ and note that according to the
above construction it is natural to have a dual vertex for each face touching
the boundary. We embed the dual such that each dual vertex is in the
corresponding face, dual pairs of edges intersect exactly once and other pairs
of edges do not intersect away from their endpoints. The superposition graph
of $G^{\\#\delta}$ and its dual is defined as the graph whose vertex set is
the union of all primal vertices, dual vertices, and intersection points
between primal and dual edges and whose edge set is the set of “half edges”
connecting a primal/dual vertex to an intersection vertex. We see it as a
weighted graph by giving weight 1 to all half dual edges and weight $p(x\to
y)$ to the half edge from a primal vertex $x$ to the intersection along the
edge $x\to 1$. We also call reference dual vertex the vertex adjacent to the
part of the boundary just after the reference prime end.

######  Theorem 2.23.

There is a measure preserving bijection between the uniform spanning tree
measure on $D^{\\#\delta}$ and the dimer measure on the superposition graph
with the boundary primal vertices and the reference dual vertex removed.

There is also a way to read the height function from the tree in this
bijection. For this we need to introduce the winding field of a tree. For each
face of the superposition graph, choose a smooth path connecting the adjacent
primal and dual vertices and choose a marked point inside this face. We call
these paths diagonals. For each face $f$, we call $\gamma_{f}$ the path
obtained by starting at the marked point, following the diagonal up to the
primal vertex, then following the UST branch from that vertex to $\partial D$,
then the path $\partial D$ up to $x_{\text{ref}}$ and then
$\gamma_{\text{ref}}$. The winding field of the UST is then by definition the
function $f\to W(\gamma_{f},\gamma_{f}(0))$.

######  Proposition 2.24.

There exist a choice of reference flow in the definition of the height
function such that, in the bijection of Theorem 2.23, the height function of a
dimer configuration is equal to the winding field of the corresponding tree
divided by $2\pi$.

######  Remark 2.25.

Since $\gamma_{f}$ in the above construction is always a path going to
infinity, the winding of $\gamma_{f}$ around its endpoint must be $0$ so
$W(\gamma_{f},\gamma_{f}(0))$ is the natural generalisation to possibly rough
curves of the intrinsic winding of $\gamma_{f}$.

The second map between uniform spanning trees and the dimer model occurs on a
special type of graphs called T-graphs and its construction is more involved
than the above one so we will not give any detail. For our purpose, it is
enough to point out that there is a direct analogue of Theorems 2.23 and 2.24.
Crucially it is still true in that case that the winding field of the UST is
the height function of the associated dimer model. We refer to [BLR16] for
details.

######  Remark 2.26.

By definition of the winding, $\gamma\to W(\gamma,x)$ depends continuously on
the path $\gamma$ as long as $x\notin\gamma$ and it is not difficult to see
that $\gamma\to W(\gamma,\gamma(0))$ also changes continuously when $\gamma$
is moved away from its starting point. On the other hand thanks to our
convention that $\gamma_{{\text{ref}}}$ merges with the horizontal axis at
some point, we see that for any face $f$,
$W(\gamma_{f},\gamma_{f}(0))\equiv-\arg\gamma_{f}^{\prime}(0)[2\pi]$ which
actually does not depend on the spanning tree. Indeed $\gamma_{f}^{\prime}(0)$
is determined only by the choice of diagonal inside $f$. Combining these two
observations, we obtain that $W(\gamma_{f},\gamma_{f}(0))$ is invariant under
small modifications of $\gamma_{f}$ away from $f$.

##  3 Upper bound on $\frac{d\nu_{1}}{d\nu_{2}}$

We now start the proof of Theorem 1.1. Recall from the introduction that we
consider three simply-connected domains $U,D_{1},D_{2}$ with
$\overline{U}\subset D_{1}\cap D_{1}$ and that $\mu_{1}=\mu_{1}^{\\#\delta}$
and $\nu_{1}=\nu_{1}^{\\#\delta}$ denote respectively the law of UST in
$D_{1}^{\\#\delta}$ and its marginal obtained by only looking at the
restriction of the UST to $U$, with similar notations for $D_{2}^{\\#\delta}$.
From now on and until the end of Section 4, we fix two such domains $D_{1}$
and $D_{2}$. We can assume without loss of generality that $D_{1}\subset
D_{2}$ since otherwise we can just compare each domain to $D_{1}\cup D_{2}$
and we also first assume that $D_{1}$ and $D_{2}$ have smooth boundaries and
defer to Corollary 5.1 the extension to domains with rough boundaries.

With these assumptions in mind, we start in this section with the first bound
under the $\nu_{1}$ measure from Theorem 1.1 where the upper bound is actually
the only non-trivial part.

###  3.1 Coupling

The upper bound $\nu_{1}(\frac{d\nu_{1}}{d\nu_{2}}\leq C)\geq\epsilon$ means
that any event (on $U$) which can happen in the smaller domain $D_{1}$ can
also happen with a somewhat similar probability in the bigger domain $D_{2}$.
To prove such a statement, the main step will be to couple $\mu_{1}$ with a
(not too strongly) biased version of $\mu_{2}$.

######  Proposition 3.1.

For all $\epsilon>0$, there exists $p>0$ such that for all $\delta$ small
enough, there exists an event $E^{\\#\delta}$ such that

$d_{TV}\Big{(}\nu^{\\#\delta}_{1},\nu^{\\#\delta}_{2}(.|E^{\\#\delta})\Big{)}\leq\epsilon\qquad\text{and}\qquad\mu^{\\#\delta}_{2}(E^{\\#\delta})\geq
p.$

With a slight abuse of notation, in the above statement we denote by
$\nu^{\\#\delta}_{2}(.|E^{\\#\delta})$ the law of the restriction to $U$ of
$\mu^{\\#\delta}_{2}(.|E^{\\#\delta})$ even if $E^{\\#\delta}$ is not
measurable with respect to this restriction.

The first step of the proof is to define the event $E^{\\#\delta}$. Recall
that $D_{1}$ is a locally connected set so $\partial D_{1}$ can be
parametrised by a curve, which we still call $\partial D_{1}$ with a slight
abuse of notation. Fix $\delta>0$ and take $x_{0}\in D_{2}^{\\#\delta}$ such
that $|\partial D_{1}(0)-x_{0}|<r$ and let $Y^{0}$ denote the branch in the
UST of $D_{2}$ starting at $x_{0}$ (parametrised in $[0,1]$ so that
$Y^{0}(1)\in\partial D_{2}^{\\#\delta}$). For any $r>0$ (to be chosen small
later), let $E_{r}$ be the event defined as follows : there exists $t>0$ such
that

  * •

As curves up to re-parametrisation, $Y^{0}([0,t])$ and $\partial D_{1}$ are at
distance at most $r$,

  * •

$Y^{0}[t,1]\subset D_{2}^{\\#\delta}\setminus D_{1}^{\\#\delta}$.

By Lemmas 2.3 and 2.4, $\mathbb{P}(E_{r})>p$ for some $p>0$ depending on $r$
but not of $\delta$ (as long as it is small enough).

We now introduce a coupling of $\mu_{1}$ and $\mu_{2}(.|E_{r})$ as follows.
First we sample the branch from $x_{0}$ conditioned on $E_{r}$. Then we sample
the rest of the trees by Wilson’s algorithm using the same sequence of
starting points $x_{1},x_{2},\ldots$ for the walks in both domains. More
precisely, for each starting point $x_{j}$ we consider an independent random
walk $X^{j}$ up to its first exist of $D_{2}$ (which also determines of course
the walk up to its exit from $D_{1}$). We can use these walks to build either
a spanning tree of $D_{1}^{\\#\delta}$ or $D_{2}^{\\#\delta}$ by just
considering different stopping times so globally we obtain a coupling of
$\mu_{1}$ and $\mu_{2}(.|E_{r})$. In other word, we see Wilson’s algorithm in
any domain as a function of random walk trajectories and we just apply two of
these functions to the same set of trajectories. Let us denote by
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ the UST of $D_{1}^{\\#\delta}$ and
$D_{2}^{\\#\delta}$ respectively under this coupling. We will denote by
$\mathcal{T}_{i}^{j}$ the partial tree generated by the first $j$ branches in
$D_{i}$, with the convention that $\mathcal{T}_{1}^{0}=\partial
D_{1}^{\\#\delta}$ and $\mathcal{T}_{2}^{0}$ is the union of $\partial
D_{2}^{\\#\delta}$ and the branch from $x_{0}$. Similarly we let $Y_{i}^{j}$
denote the loop erasure of the $j$th walk in the algorithm in $D_{i}$, i.e
$Y_{i}^{j}=\mathcal{T}_{i}^{j}\setminus\mathcal{T}_{i}^{j-1}$.

######  Lemma 3.2.

For all $\epsilon>0$, there exists $C>0$ such that for all $n\geq 1$, for all
$r>0$ small enough and $\delta$ small enough, for all sequence of points
$x_{j}\in D_{1}^{\\#\delta}\cap D_{2}^{\\#\delta}$ the above coupling
satisfies

$\mathbb{P}(\mathcal{T}^{n}_{1}1_{\tilde{D}_{C^{n}r}}=\mathcal{T}^{n}_{2}1_{\tilde{D}_{C^{n}r}})\geq
1-n\epsilon$

where $\tilde{D}_{r}=\\{x\in D_{1}:d(x,\partial D_{1})\geq r\\}$.

###### Proof.

We proceed by induction. First for $n=1$, let $\tau$ be the first time where
$X^{1}$ hits $\mathcal{T}_{1}^{0}\cup\mathcal{T}_{2}^{0}$, evidently the loop-
erasures agree up to that time. Suppose first that
$X^{1}(\tau)\in\mathcal{T}_{1}^{0}$. By definition of the event $E_{r}$,
$X^{1}(\tau)$ is within distance $r$ of $\mathcal{T}_{2}^{0}$. By Beurling’s
estimate (Lemma 2.4), we can find $C$ large enough (and independent of
everything else) so that with probability $1-\epsilon$, the random walk will
hit $\mathcal{T}_{2}^{0}$ without exiting $B(X^{1}(\tau),Cr)$. Note that this
also shows that with probability $1-\epsilon$, after the first exit from
$\tilde{D}_{Cr}$ the walk does not come back to $\tilde{D}_{C^{2}r}$. When
both events occur, any loop erased after $\tau$ must have a point outside
$\tilde{D}_{Cr}$ but any such loop cannot exit $\tilde{D}_{C^{2}r}$ and of
course any piece of the loop-erased walk added after $\tau$ is in
$B(X^{1}(\tau),Cr)$, in particular we have
$\mathcal{T}_{1}^{1}1_{\tilde{D}_{C^{2}r}}=\mathcal{T}^{1}_{2}1_{\tilde{D}_{C^{2}r}}$
as desired (with $2\epsilon$ and $C^{2}$).

Now consider the case where $X^{1}(\tau)\in\mathcal{T}_{2}^{0}$, and note that
we must have $X^{1}(\tau)\in Y^{0}\cap D_{1}^{\\#\delta}$ since by definition
$X^{1}$ has not yet exited from $D_{1}^{\\#\delta}$. Again by definition of
$E_{r}$, $X^{1}(\tau)$ must be within distance $r$ from $\partial D_{1}$ so we
are back in the same situation as above.

For $n>1$, we argue exactly as for the $n=1$ case. The only difference is that
if $X^{n}$ hits $\mathcal{T}^{n-1}$ before exiting $\tilde{D}_{C^{n-1}r}$
there is nothing to prove. ∎

We are now ready to finish the proof of the proposition.

###### Proof of Proposition 3.1.

Fix $\epsilon_{0}>0$ and $\eta>0$ such that for all $x\in U$ and $y\in\partial
D_{1},d(x,y)>2\eta$.

By the finiteness theorem 2.14, we can find $n$ and $x_{1},\ldots x_{n}$ such
that, with probability $1-\epsilon_{0}/2$, after the first $n$ steps no
remaining random walk used in Wilson’s algorithm will reach distance $\eta$
from $U$.

We then apply Lemma 3.2 with $\epsilon=\epsilon_{0}/2n$ and $r$ sufficiently
small so that $C^{n}r<\eta$. Lemma 3.2 shows that
$\mathbb{P}(\mathcal{T}^{n}_{1}1_{\tilde{D}_{\eta}}=\mathcal{T}^{n}_{2}1_{\tilde{D}_{\eta}}|E_{r})\geq
1-\epsilon_{0}/2$. Recalling the finiteness theorem, this shows that
$\mathbb{P}(\mathcal{T}_{1}1_{U}=\mathcal{T}_{2}1_{U}|E_{r})\geq
1-\epsilon_{0}$ as desired since as noted just after its definition the
probability of $E_{r}$ can be bounded independently of $\delta$. ∎

###  3.2 Radon-Nikodym bound

We now derive from the coupling of the previous section a bound of the form
used in Theorem 1.1.

######  Lemma 3.3.

Let $\nu_{1},\nu_{2},\tilde{\nu}_{2}$ be three measures such that, for some
$\epsilon,p>0$,

$d_{TV}(\nu_{1},\tilde{\nu}_{2})\leq\epsilon,\quad\text{and}\quad\frac{d\tilde{\nu}_{2}}{d\nu_{2}}\leq\frac{1}{p}.$

Then we have

$\nu_{1}(\epsilon\leq\frac{d\nu_{1}}{d\nu_{2}}\leq\frac{2}{p})\geq
1-5\epsilon.$

###### Proof.

First the lower bound is trivial, indeed

$\nu_{1}(\frac{d\nu_{1}}{d\nu_{2}}<\epsilon)=\int\frac{d\nu_{1}}{d\nu_{2}}1_{\frac{d\nu_{1}}{d\nu_{2}}<\epsilon}d\nu_{2}\leq\epsilon.$

For the upper bound, we first note that interpreting the total variation
distance as the $L^{1}$ norm on densities, we get

$\int\lvert\frac{d\tilde{\nu}_{2}}{d\nu_{1}}-1\rvert d\nu_{1}\leq 2\epsilon.$

By Markov,
$\nu_{1}(\lvert\frac{d\tilde{\nu}_{2}}{d\nu_{1}}-1\rvert\geq\frac{1}{2})\leq
4\epsilon$ so $\nu_{1}(\frac{d\tilde{\nu}_{2}}{d\nu_{1}}\leq\frac{1}{2})\leq
4\epsilon$, or in other words

$\nu_{1}(\frac{d\nu_{1}}{d\tilde{\nu}_{2}}\leq 2)\geq 1-4\epsilon.$

We conclude by the assumption
$\frac{d\tilde{\nu}_{2}}{d\nu_{2}}\leq\frac{1}{p}$ and union bound. ∎

Applying Lemma 3.3 with $\tilde{\nu}_{2}=\nu_{2}(.|E)$ for the event $E$ given
by Proposition 3.1, we immediately obtain the first half of Theorem 1.1:

######  Proposition 3.4.

For any $\epsilon>0$ there exists $C>0$ such that for all $\delta$ small
enough,

$\nu_{1}(\frac{1}{C}\leq\frac{d\nu_{1}}{d\nu_{2}}\leq C)\geq 1-\epsilon.$

This last statement can be also rewritten as an inequality involving the
probabilities under both measures for any event.

######  Proposition 3.5.

There exists two increasing functions $f$ and $g$ from $(0,1)$ to itself such
that for any event $A$, we have the following inequality,

$g(\nu_{1}(A))\leq\nu_{2}(A)\leq f(\nu_{1}(A)).$

###### Proof.

For $\epsilon>0$, let $C_{\epsilon}$ be the constant from Proposition 4.11 and
let $A$ be any event. We define the event
$E_{\epsilon}:=\\{\frac{1}{C_{\epsilon}}\leq\frac{d\nu_{1}}{d\nu_{2}}\leq
C_{\epsilon}\\}$ where $C$ is as in Proposition 3.4. By union bound we have
$\nu_{1}(A)\leq\epsilon+\nu_{1}(A)$ which leads us to the inequality
$\nu_{1}(A)\leq\epsilon C_{\epsilon}\nu_{2}(A)$ by Proposition 3.4.
Defining $G(x):=\inf_{\epsilon}\\{\epsilon+C_{\epsilon}x\\}$ and
$g(x):=\inf\\{u,G(u)\geq x\\}$ we have the first half of the inequality,

$g(\nu_{1}(A))\leq\nu_{2}(A)$

In a similar fashion, we have by union bound and Proposition 3.4 the
upperbound $\nu_{2}(A)\leq\nu_{2}(E_{\epsilon}^{c})+C_{\epsilon}\nu_{1}(A)$.
Leading us to the second part of the inequality

$\nu_{2}(A)\leq f(\nu_{1}(A))$

in which $f(x):=\inf_{\epsilon}\\{\nu_{2}(E_{\epsilon}^{c})+C_{\epsilon}x\\}$
∎

##  4 Lower bound on $\frac{d\nu_{1}}{d\nu_{2}}$

We now turn to the second inequality from Theorem 1.1. As in the previous
section the main point is the following coupling statement.

######  Proposition 4.1.

For all $\epsilon>0$, there exists $p>0$ depending only on $U,D_{1},D_{2}$
such that for all $\delta$ small enough, there exists a law $\tilde{\nu}_{1}$
such that

$d_{TV}(\nu_{2},\tilde{\nu}_{1})\leq\epsilon,\qquad\frac{d\tilde{\nu}_{1}}{d\nu_{1}}\leq\frac{1}{p}.$

We think of this proposition as an analogue of Proposition 3.1 because we see
$\tilde{\nu}_{1}$ as a kind of conditioned version of $\nu_{1}$, with a
conditioning event of probability at least $p$. This case is however
significantly more involved than the previous one so $\tilde{\nu}_{1}$ is not
exactly obtained by a conditioning.

Let us highlight the main steps of the proof. As before, we first focus on a
single branch. The challenge compared to the previous section is that is no
longer possible to simply turn the larger domain into a close approximation of
the smaller one by an appropriate event. Instead, given a random walk in the
large domain $D_{2}$, we want to create a path in $D_{1}$ which agrees with
the first one on $U$ (so that their loop-erasure match) and reproduces the
same topology with its excursions outside of $U$ (so that any erasure coming
from the excursions outside of $U$ matches).

Figure 3: An example where two coupled trajectories lead to different loop-
erasures. Consider the two paths obtained by following one of dashed colored
lines followed by the continuous black line. The points where the paths come
very close to itself should be though as actual intersection and are only
drawn with a small gap for readability. The loop erasures of the path starting
red is the upper straight line while for the blue path it is the bottom
straight line. Note that in both case, all erased loops are bounded.

The first step is to clarify how to go from a coupling of two random walks to
their loop-erasures (see Figure 3 for why this is non-trivial). Here the main
idea is to introduce a loop-erasing procedure with both forward and backward
aspects, designed to ensure that the loop-erasure are obtained “locally” from
the random walks and must therefore agree whenever the walks agree.

The second step is actually a trick to get rid of all the topological
complexity from possible large scale loop erasure : By Lemma 2.10, the
conditional law of a random walk $X$ given that its loop erasure is some
simple path $\gamma$ is obtained by adding to $\gamma$ loops sampled from an
independent loop soup. By Lemma 2.12, with positive probability all these
added loops are small. On that even all random walks are guarantied to remain
within a small distance of their erasures and no large scale complexity can
happen from the erasing procedure.

Finally, as in Section 3, we will focus on the first loop-erased walk in
Wilson’s algorithm. The proof for any finite number will be similar and an
application of Schramm’s finiteness theorem will conclude the proof by going
from a finite number of branches to the full tree.

###  4.1 Mixed Loop-erasing

Denote by $\operatorname{\overrightarrow{LE}}(X)$ and
$\operatorname{\overleftarrow{LE}}(X)$ the forward and backward loop-erasure
of a path $X$ and by $\oplus$ the concatenation of paths.

######  Definition 4.2.

Let $X=(X_{t})_{0\leq t\leq T_{i_{\max}}}$ and $(T_{i})_{1\leq i\leq i_{max}}$
be respectively a path and a sequence of times with $T_{i}<T_{i+1}$ almost
surely. We define the mixed loop erasure of $X$ with respect to the sequence
$T_{i}$ by the following inductive procedure (see Figure 4 for an
illustration):

  * •

We define $Y_{1}:=\operatorname{\overleftarrow{LE}}(X[0,T_{1}])$.

  * •

Given $Y_{i}$ a simple path, let $s_{i}=\inf\\{s:Y_{i}(s)\in
X[T_{i},T_{i+1}]\\}$ and let
$t_{i}=\max\\{t\in[T_{i},T_{i+1}]:X(t)=Y(s_{i})\\}$. In other words,
$Y_{i}(s_{i})$ is the first point where $Y_{i}$ intersect the trajectory of
$X$ and $t_{i}$ is the (last) time in $X$ where this intersection occurs.

  * •

We set
$Y_{i+1}:=Y_{i}[0,s_{i}]\oplus\operatorname{\overleftarrow{LE}}(X[t_{i},T_{i+1}])$.

  * •

we define the mixed loop-erasure to be the final simple path $Y_{i_{\max}}$.

Figure 4: One step in the mixed loop-erasing procedure. Left : the “current”
loop erasure $Y_{i}$ in dark complete line and the additional portion of walk
$X[t_{i},T_{i+1}]$ in dashed red. Center : $Y[0,s_{i}]$ and
$X[t_{i},T_{i+1}]$; in a sense the forward loop from $Y_{i}(s_{i})$ to
$X(t_{i})$ has been erased. Right : $Y_{i+1}$ obtained by doing a reverse
loop-erasure. Note that in this example, the output of the mixed loop erasure
is different from either the forward or backward one.

It is well known that the forward and backward loop-erasure of a random walk
have the same law. For the mixed loop-erasing, we cannot give such a general
statement but we nonetheless have the following.

######  Lemma 4.3.

Let $G$ denote a graph with a boundary $\partial G$ and let $X$ denote a
random walk stopped when it first hits $\partial G$. Let $E_{i}$ be a sequence
of subsets of $G$, let $T_{i}$ be the sequence of stopping times defined by
$T_{i+1}=\inf\\{t>T_{i}:X_{t}\in E_{i+1}\cup\partial G\\}$ and let $i_{\max}$
be the first $i$ such that $T_{i}\in\partial G$, i.e
$T_{i_{\max}}=\inf\\{t:X_{t}\in\partial G\\}$ is the time where we stop the
walk. The mixed loop erasure of $X$ with respect to the times $T_{i}$ of $X$
has the law of the (forward) loop erasure of $X$.

###### Proof.

We claim that the sequence of paths $(Y_{i})_{0\leq i\leq i_{\max}}$ is equal
in law the the sequence
$(\operatorname{\overrightarrow{LE}}(X[0,T_{i}]))_{0\leq i\leq i_{\max}}$.
Indeed suppose by induction that this is true for some $i$, fix some path
$\gamma$, and let us compare the law of $Y_{i+1}$ given $Y_{i}=\gamma$ to the
law of $\operatorname{\overrightarrow{LE}}(X[0,T_{i+1}])$ given
$\operatorname{\overrightarrow{LE}}(X[0,T_{i}])=\gamma$. Also define
$s_{i}=\inf\\{s:\gamma(s)\in X[T_{i},T_{i+1}]\\}$ and
$t_{i}=\max\\{t\in[T_{i},T_{i+1}]:X(t)=\gamma(s_{i})\\}$ following the
notation from Definition 4.2.

Note that
$\operatorname{\overrightarrow{LE}}(X[0,T_{i+1}])=\gamma[0,s_{i}]\oplus\operatorname{\overrightarrow{LE}}(X[t_{i},T_{i+1}])$
and that, conditionally on $s_{i}$, $X[t_{i},T_{i+1}]$ is a random walk
conditioned to hit $E_{i+1}$ before $\gamma$. By Lemma 7.2.1 in [Law12], there
is a measure-preserving bijection $\varphi_{i}$ on such excursions paths
satisfying

$\operatorname{\overleftarrow{LE}}(\varphi_{i}(X[t_{i},T_{i+1}]))=\operatorname{\overrightarrow{LE}}(X[t_{i},T_{i+1}]).$

Note that this includes the fact that $t_{i}$ and $T_{i+1}$ are invariant
under $\varphi_{i}$. In particular

$Y_{i+1}:=\gamma[0,s_{i}]\oplus\operatorname{\overleftarrow{LE}}(X[t_{i},T_{i+1}])\overset{\text{law}}{=}\gamma[0,s_{i}]\oplus\operatorname{\overrightarrow{LE}}(X[t_{i},T_{i+1}])=\operatorname{\overrightarrow{LE}}(X[0,T_{i+1}]).$

∎

From the proof, we also have the following corollary

######  Corollary 4.4.

Given sets $E_{i}$ (with associated hitting times $T_{i}$) as above, there
exists a bijective, measure preserving, map $\varphi$ from the set of random
walks trajectory to itself such that the forward loop-erasure of $X$ and the
mixed loop-erasure of $\varphi(X)$ with respect to the times $T_{i}$ are
identical.

With a slight abuse of terminology, we will speak in the following of the
mixed loop-erasure with respect to a sequence of sets in the above context.

###  4.2 RW in $D_{2}$

In this section, as mentioned in the beginning of the section, we turn to the
construction of a RW path $\tilde{X}_{2}$ whose mixed loop-erasure has exactly
the law of a branch of UST in $D_{2}$ and yet never creates any large loop. As
an aside on notations, we will use unmarked letters to describe objects with
the original law of random walk/UST/…and a tilde to denote biased versions
used to construct the measure $\tilde{\nu}_{1}$. Without loss of generality,
we will start our walk at $0$ to simplify notations.

From now on we fix three open connected sets $U_{1},U_{2},U_{3}$ with
$\overline{U}\subset U_{1}$, $\overline{U_{i}}\subset U_{i+1}$ and
$\overline{U}_{3}\subset D_{1}$ and we let $E_{i}$ denote the alternating
sequence of sets $E_{2i}=U_{2}^{\\#\delta}$ and $E_{2i+1}=D_{1}\setminus
U_{3}^{\\#\delta}$. Fix $r$ to be chosen small enough later (in particular so
that the boundaries of $U,U_{1},U_{2},U_{3},D_{1},D_{2}$ are all at distance
more than $r$) and we consider the event $G=G_{r}$ that all loops in the loop
soup $(\ell_{k})_{k\geq 1}$ have diameter at most $r$. Let $Y_{2}$ be a loop-
erased random walk from $0$ in $D_{2}^{\\#\delta}$, we define $X_{2}$ and
$\tilde{X}_{2}$ by sampling respectively an unbiased loop soup or one
conditioned on $G_{r}$ independent from $Y_{2}$, then applying 2.9 and finally
applying the bijection of Corollary 4.4 relative to the sequence of sets
$E_{1}$.

######  Lemma 4.5.

$Y_{2}$ is the mixed loop erasures of both $X_{2}$ and $\tilde{X}_{2}$ with
respect to the sets $E_{i}$. Furthermore the pair $(X_{2},Y_{2})$ has the law
of a random walk and its loop-erasure. Finally there exists $C$ depending only
on $D_{2}$ such that for all $\delta$ small enough and for all paths $\chi$ on
$D_{2}^{\\#\delta}$

$\mathbb{P}(\tilde{X}^{2}=\chi|Y_{2})\leq C\mathbb{P}(X^{2}=\chi|Y_{2}).$

###### Proof.

The first two points are trivial by construction and because the map from
Corollary 4.4 is a measure preserving bijection. The last point is because by
Lemma 2.12, the even $G_{r}$ has a strictly positive probability independently
of $\delta$. ∎

Finally, as mentioned above the reason why we introduce the conditioning by
$G_{r}$ is to get rid of possible large scale erasures. For this we want to
say that in a sense the $r$ neighbourhood of $\chi$ is “simple” with high
probability.

######  Definition 4.6 (Definition 3.3 in [Sch00]).

An $r,R$-quasiloop of a path $\chi$ is a pair $a,b$ such that
$\lvert\chi(a)-\chi(b)\rvert\leq r$ but $\operatorname{diam}(\chi[a,b])\geq
R$.

######  Lemma 4.7 (Lemma 3.4 in [Sch00]).

For all sequence of starting points, for all $R>0$

$\lim_{r\to 0}\limsup_{\delta\to 0}\mathbb{P}(Y\text{ has an
$r,R$-quasiloop})=0.$

###  4.3 RW in $D_{1}$

In this section we define the walk in $D_{1}$ and control that its law is not
too far from that of a simple random walk. Note that the construction will
depend on $U_{1},U_{2},U_{3}$ which were previously defined, the distance $r$
that was used to define $X_{2}$ and an integer $N$. We also fix a
diffeomorphism $\phi$ from $D_{2}$ to $D_{1}$ such that the restriction of
$\phi$ to $U_{3}$ is the identity.

Define a sequence of times $T_{i}=T_{i}^{2}$ inductively by $T_{0}=0$ and
$T_{2i+1}=\inf\\{t\geq T_{2i},X^{2}(t)\not\in U_{3}\\}$, $T_{2i}=\inf\\{t\geq
T_{2i-1}:X^{2}(t)\in U_{2}\\}$. We also let $T_{\max}$ denote the hitting time
of $\partial D_{2}$ by $X^{2}$ and we let
$i_{\max}=\max\\{i:T_{i}<T_{\max}\\}$ i.e $i_{\max}$ is the last time in the
sequence where we are on $\partial U_{3}$. We will denote by $T^{1}$ and
$T^{Y}$ the corresponding set of times for either $X^{1}$ (when it will be
defined) and $Y_{2}$.

We say that the path $Y_{2}$ is _$\epsilon$ -good_ is it satisfies the
following:

  * •

It does not have any $(\epsilon^{2},\epsilon)$-quasiloop;

  * •

$i_{\max}^{Y}\leq 1/\epsilon$;

  * •

For every $i$, let $T_{2i^{-}}^{Y}$ be the first time after $T_{2i-1}^{Y}$
where $Y$ comes within distance $\epsilon^{2}$ of $U_{2}$, we want that
$Y[T_{2i^{-}},T_{2i}]\subset B(Y(T_{2i^{-}}),\epsilon)$.

Combining the probability to see quasi-loops from Lemma 4.7, the Beurling
estimate from Lemma 2.4 and the fact that by uniform crossing $i_{\max}$ has
an exponential tail, we see that the probability that $Y_{2}$ is
$\epsilon$-good goes to $1$ as $\epsilon\to 0$, uniformly over the mesh size
$\delta$.

We define the path $\tilde{X}^{1}$ conditionally on $\tilde{X}^{2}$ (and
therefore $Y^{2}$) by concatenating different pieces of random walk and recall
that we denote by $\tilde{X}[t,t^{\prime}]$ the subpath between times $t$ and
$t^{\prime}$.

  * •

If $Y_{2}$ is not $\epsilon$-good, then we just let $\tilde{X}^{1}$ be a
random walk independent of everything else. From now on assume that $Y_{2}$ is
$\epsilon$-good and for ease of notation assume that $i_{\max}\geq 3$.

  * •

We set $\tilde{X}^{1}[0,T_{1}^{1}]:=\tilde{X}^{2}[0,T_{1}^{2}]$.

  * •

We choose a sequence of rectangle following
$\phi(\tilde{X}_{2}[T_{1}^{2},T_{2}^{2}])$ up to a precision $r$ (see Section
2.1 ) and such that only the last one comes within distance $\epsilon^{2}/2$
of $U_{2}$. We let $\tilde{X}^{1}[T_{1}^{1},T_{2}^{1}]$ be a random walk
started from $\tilde{X}^{1}(T_{1}^{1})=\tilde{X}^{2}(T_{1}^{2})$, conditioned
to first hit $U_{2}$ at the position $\tilde{X}^{2}(T_{2}^{2})$ and to cross
all the rectangles in the above sequence in order.

  * •

We set
$\tilde{X}^{1}(T_{2}^{1},T_{3}^{1}):=\tilde{X}^{2}(T_{2}^{2},T_{3}^{2})$ which
is possible since by construction
$\tilde{X}_{2}(T^{2}_{2})=\tilde{X}^{1}(T_{2}^{1})$.

  * •

We iterate the construction up to $i_{\max}$. From $T_{i\max}$ we complete
$\tilde{X}^{1}$ by a random walk following
$\phi(\tilde{X}_{2}[T_{i_{\max}},T_{\max}])$ and exiting $D_{1}$.

######  Lemma 4.8.

For all $\epsilon>0$ and $r>0$ such that $r\leq\epsilon^{2}$, there exists
some $C>0$ such that for all $\chi$

$\mathbb{P}(\tilde{X}^{1}=\chi)\leq C\mathbb{P}(X^{1}=\chi)$

where $X^{1}$ is a random walk stopped when it exits $D_{1}$.

###### Proof.

First by construction, if $Y_{2}$ is not $\epsilon$-good there is nothing to
prove so let us analyse $\mathbb{P}(\tilde{X}_{1}=\chi|Y_{2})$ in the case
where $Y_{2}$ is $\epsilon$-good. To construct $\tilde{X}_{1}$, we need to
know $Y_{2}$ and all the pieces of trajectory $\tilde{X}^{2}[T_{2k},T_{2k+1}]$
but conditionally on that information we sample independent random walks,
therefore

$\mathbb{P}(\tilde{X}_{1}=\chi|Y_{2})=\mathbb{P}\Big{(}\forall
k<\tfrac{i_{\max}}{2},\,\tilde{X}_{2}[T_{2k},T_{2k+1}]=\chi[T_{2k},T_{2k+1}]\Big{|}Y_{2}\Big{)}\\\
\times\prod_{k\leq\frac{i_{\max}}{2}}\mathbb{P}_{\chi(T_{2k-1})}\Big{(}X[0,T]=\chi[T_{2k-1},T_{2k}]\Big{|}X(T)=\chi(T_{2k}),\text{crossing
rectangles},Y_{2}\Big{)}\\\
\times\mathbb{P}_{\chi(T_{i_{max}})}\Big{(}X[0,T]=\chi[T_{i_{\max}},T_{\max}]\Big{|}\text{crossing
rectangles},Y_{2}\Big{)},$

where in the second line we denote by $X$ a simple random walk started at the
position $\chi(T_{2k-1})$ indicated in the probability and by $T$ the hitting
time of $U_{2}$ while the conditioning event is given in the third step in the
definition of $\tilde{X}_{1}$. In the third line we use similar notations
referring to the last step.

We start by analysing the terms in the second line, i.e from $2k-1$ to $2k$.
By the lack of $(\epsilon^{2},\epsilon)$-quasi loop in $Y$, we can find an
uniform bound on the number of rectangles that have to be crossed to follow
$\phi(Y)$. By the conditional uniform crossing, each rectangle before the last
has uniformly positive probability to be crossed by the random walk (since
they are all at distance at least $\epsilon^{2}/2$ of $\partial U_{2}$).
Therefore, up to a multiplicative factor, we can forget about the conditioning
to cross rectangles in the second line but note that this also removes any
dependence on $Y$. For the last step (i.e the third line in the equation), by
uniform continuity of $\phi$ we have a uniform bound on the number of
rectangles that have to be crossed to follow $\phi(Y)$ from
$\chi(T_{i_{\max}})$ up to a very small distance of $\partial D_{1}$ and from
there Lemma 2.4 shows that the walk is likely to exit $D_{1}$ without going
far away. Overall if we also integrate to remove the conditioning on $Y_{2}$,
we obtain

$\mathbb{P}(\tilde{X}_{1}=\chi)\leq
C\mathbb{E}_{Y_{2}}\Big{[}\mathbb{P}(\forall
k<i_{\max}/2,\tilde{X}_{2}[T_{2k},T_{2k+1}]=\chi[T_{2k},T_{2k+1}]\mid
Y_{2})\Big{]}\\\
\times\prod_{k\leq\frac{i_{\max}}{2}}\mathbb{P}_{\chi(T_{2k-1})}\Big{(}X[0,T]=\chi[T_{2k-1},T_{2k}]\Big{|}X(T)=\chi(T_{2k})\Big{)}\\\
\times\mathbb{P}_{\chi(T_{i_{max}})}\Big{(}X[0,T]=\chi[T_{i_{\max}},T_{\max}]\Big{)}\Big{)},$

By Lemma 4.5 when conditioned on $Y_{2}$, the law of $\tilde{X}_{2}$ is
absolutely continuous with respect to a random walk conditioned by its loop-
erasure. When taking the expectation over $Y_{2}$, we therefore recover up to
constant the law of a random walk, i.e we have

$\mathbb{P}(X_{1}=\chi)\leq C\mathbb{P}(\forall
k<i_{\max}/2,X[T_{2k},T_{2k+1}]=\chi[T_{2k},T_{2k+1}])\\\
\times\prod_{k\leq\frac{i_{\max}}{2}}\mathbb{P}_{\chi(T_{2k-1})}\Big{(}X[0,T]=\chi[T_{2k-1},T_{2k}]\Big{|}X(T)=\chi(T_{2k})\Big{)}\\\
\times\mathbb{P}_{\chi(T_{i_{max}})}\Big{(}X[0,T]=\chi[T_{i_{\max}},T_{\max}]\Big{)}.$

Now for the random walk, it is not hard to see that

$\mathbb{P}\Big{(}\forall
k<i_{\max}/2,X[T_{2k},T_{2k+1}]=\chi[T_{2k},T_{2k+1}]\Big{)}\leq
C\textrm{harm}_{D_{2}\setminus U_{2}}(\chi(T_{i_{\max}}),\partial D_{2})\\\
\times\prod\mathbb{P}_{\chi(T_{2k})}\Big{(}X[0,T]=\chi[T_{2k},T_{2k+1}]\Big{)}\textrm{harm}_{D_{2}\setminus
U_{2}}(\chi(T_{2k-1}),\chi(T_{2k})),$

where $\textrm{harm}_{D_{2}\setminus U_{2}}(\chi(T_{2k-1}),\chi(T_{2k}))$ is
the probability starting from $\chi(T_{2k-1})$ to reach $U_{2}$ before exiting
$D_{2}$ and to do so through $\chi(T_{2k})$. When combining the previous two
equations, the harmonic measure in $D_{1}$ and $D_{2}$ are within a constant
of each other and they compensate the conditioning on the endpoints for odd
intervals. We conclude by comparing the expression for simple random walk and
for $X_{1}$. ∎

###  4.4 loop-erasure

Now that we have our candidate random walk in $D_{1}$, we can look at its
loop-erasure. Let $\tilde{Y}_{1}$ denote the mixed loop erasure of
$\tilde{X}_{1}$ with respects to the alternating sequence of sets
$E_{2i}=U_{2}^{\\#\delta}$ and $E_{2i+1}=D_{1}^{\\#\delta}\setminus
U_{3}^{\\#\delta}$.

######  Lemma 4.9.

There exists $C>0$ such that for all $\epsilon>0$ if $r$ is small enough
compared to $\epsilon$ and if $Y_{2}$ is $\epsilon$-good then we have

$d(\tilde{Y}_{1},\phi(Y_{2}))\leq C\epsilon,\text{ and }\tilde{Y}_{1}\cap
U_{1}=Y_{2}\cap U_{1},$

where recall that $d(\tilde{Y}_{1},\phi(Y_{2}))$ denotes the uniform distance
up to time parametrisation.

###### Proof.

We start by proving the first statement. By construction,
$d(\tilde{X}_{2},Y_{2})\leq r$ since $\tilde{X}_{2}$ is obtained from $Y_{2}$
by adding loops of size at most $r$. Also, since $\tilde{X}_{1}$ is defined by
gluing portions of paths where $\tilde{X}_{2}=\phi(\tilde{X}_{2})$ with
portions following $\phi(X_{2})$, by construction it satisfies
$d(\tilde{X}_{1},\phi(\tilde{X}_{2}))\leq r$. Since $\phi$ is Lipschitz
(recall that it was chosen as a diffeomorphism), we therefore have
$d(\tilde{X}_{1},\phi(Y_{2}))\leq Cr$ for some $C>0$. Also using the Lipschitz
property of $\phi^{-1}$, we see that $\phi(Y_{2})$ does not have
$c\epsilon^{2},C\epsilon$ quasi-loops for some $c,C$ depending only on $\phi$.

Now let us bound the size of the loops created by $\tilde{X}_{1}$. Suppose by
contradiction that $\tilde{X}_{1}$ creates a loop of diameter $K\epsilon$ for
some $K$ to be chosen large enough later, and let $u,v,w$ be three times such
that $v\in[u,w]$, $\tilde{X}_{1}(u)=\tilde{X}_{1}(w)$ and
$d(\tilde{X}_{1}(v),\tilde{X}_{1}(u))\geq K\epsilon$. Since
$d(\tilde{X}_{1},\phi(\tilde{X}_{2}))\leq r$, there must exists corresponding
times $u^{\prime},v^{\prime},w^{\prime}$ such that
$d(\phi(Y_{2})(u^{\prime}),\phi(Y_{2}(w^{\prime}))\leq 2Cr$ and
$d(\phi(Y_{2})(u^{\prime}),\phi(Y_{2}(v^{\prime}))\leq K\epsilon-2Cr$. In
particular $\phi(Y_{2})$ must have an $(2Cr,K\epsilon-2Cr)$-quasiloop which is
a contradiction for $r$ small enough and $K$ large enough. As a corollary,
$d(\tilde{Y}_{1},\tilde{X}_{1})\leq K\epsilon$ which completes the proof of
the first point. From now on we assume that $\epsilon$ is chosen so that
$K\epsilon$ is small enough compared to the distances between the boundary of
any of the $U_{i}$ or $D_{i}$.

For the proof of the second point, let $\tilde{Y}_{1}^{i}$ denote the mixed
loop erasure of $\tilde{X}_{1}[0,T_{i}]$ and similarly for $\tilde{X}_{2}$
(this is the same notation as in the definition of the mixed loop-erasure 4.2
except that we moved the “number of step” index up because we now also have
the domain index) and let us proceed by induction. For $i=1$, there is nothing
to prove, we have $\tilde{Y}_{1}^{1}=Y_{2}^{1}$. For $i=2$ (and for general
even steps later), by the bound on the size of erased loops, we see that
$\tilde{X}_{1}[T_{1},T_{2}]$ cannot intersect the portion of $\tilde{Y}_{1}$
before its last visit to $U_{2}$. Since the same holds for $\tilde{X}_{2}$ we
must in particular have

$\tilde{Y}_{1}^{2}\cap U_{1}=\tilde{Y}_{1}^{1}\cap U_{1}=Y_{2}^{1}\cap
U_{1}=Y_{2}^{2}\cap U_{1}.$

For $i=3$ and general odd steps afterwards, let
$s_{2}^{1},s_{2}^{2},t_{2}^{1},t_{2}^{2}$ be defined as in Definition 4.2 so
that we have

$\tilde{Y}_{1}^{3}=\tilde{Y}_{1}^{2}[0,s_{2}^{1}]\oplus\operatorname{\overleftarrow{LE}}(\tilde{X}[t_{2}^{1},T_{3}]),\quad
Y_{2}^{3}=Y_{2}^{2}[0,s_{2}^{2}]\oplus\operatorname{\overleftarrow{LE}}(\tilde{X}[t_{2}^{2},T_{3}]),$

where note that crucially we do not need a subscript for the random walk
trajectory $\tilde{X}$ of the time $T_{3}$ as they refer to a portion where
$\tilde{X}_{1}=\tilde{X}_{2}$. By the bound on the size of erased loops, both
$\tilde{X}[T_{2},t_{2}^{1}]$ and $\tilde{X}[T_{2},t_{2}^{2}]$ must stay within
$2K\epsilon$ of $\tilde{X}(T_{2})$ and therefore (still using the bound on the
size of erased
loops)$\operatorname{\overleftarrow{LE}}(\tilde{X}[t_{2}^{1},T_{3}])$ and
$\operatorname{\overleftarrow{LE}}(\tilde{X}[t_{2}^{2},T_{3}])$ must agree
whenever they are at distance $3K\epsilon$ of $\tilde{X}(T_{2})$, in
particular $\tilde{Y}_{1}^{3}\cap U_{1}=Y_{2}^{3}\cap U_{1}$. We conclude the
proof by induction. ∎

###  4.5 Coupling several paths

In this section, we conclude the proof of Proposition 4.11. From Sections 4.1,
4.2, 4.2 and 4.4, we showed how to construct a single path in $D_{1}$ that
agree over $U_{1}$ with a loop erased random walk in $D_{2}$.

######  Lemma 4.10.

For all open sets $U$, $U_{1}$ such that $\bar{U}\subset U_{1}$ and
$\bar{U}_{1}\subset D_{1}$, for all homomorphism $\phi$ from $D_{2}$ to
$D_{1}$ restricting to the identity on a neighbourhood of $U_{1}$, for all
$n\geq 1$, there exits $C$ such that for all $x_{1},x_{n}\in U^{\\#\delta}$
there exists a coupling of $Y_{2}(v_{1}),\ldots,Y_{2}(v_{n})$ with a law
$\tilde{\mu}_{1}$ on paths $\tilde{Y}_{1}(v_{n}),\ldots,\tilde{Y}_{1}(v_{n})$
in $D^{\\#\delta}_{1}$ satisfying

$\displaystyle\mathbb{P}(\forall i,\,Y_{2}(v_{i})\cap
U_{1}=\tilde{Y}_{1}(v_{i})\cap U_{1})\geq 1-\epsilon,$
$\displaystyle\mathbb{P}(\forall
i,\,d(\tilde{Y}_{1}(v_{i}),\phi(Y_{2}(v_{i}))\geq\epsilon)\leq\epsilon,$
$\displaystyle\frac{d\tilde{\mu}_{1}}{d\mu_{1}}\leq C.$

where with a slight abuse of notation $\mu_{1}$ denotes the joint law of the
spanning tree branches starting from $v_{1},\ldots,v_{n}$ in $D_{1}$.

###### Proof.

We prove the result by induction on $n$. For $n=1$, we combine Lemma 4.9 to
control the first two lines on the event that $Y_{2}^{i}s$$\epsilon$-good
(whose probability goes to $1$ as $\epsilon\to 0$ with Lemma 4.8 for the third
line (the uniform bound on the Radon-Nicodym derivative there trivially
extends to the law of any function of $\tilde{X}_{1}$). Suppose now that it
holds for some $n\geq 1$. We choose $U^{(n)}_{1}$ such that
$\overline{U}_{1}\subset U^{(n)}_{1}$ and such that $\phi$ restricts to the
identity in a neighbourhood of $U_{1}^{\prime}$. By induction we can find a
law on paths $\tilde{Y}_{1}(v_{1}),\ldots,\tilde{Y}_{1}(v_{n})$ satisfying the
assumption for $U_{1}^{(n)}$ and some $\epsilon_{n}$ to be chosen small enough
later. By Wilson’s algorithm, what remains to be done is to couple loop-erased
random walks from $v_{n+1}$ in respectively
$D_{1}\setminus\cup\tilde{Y}_{1}(v_{i})$ and $D_{2}\setminus\cup
Y_{2}(v_{n})$. We can also assume without loss of generality that the
$\tilde{Y}_{1}(v_{i})$ paths and the $Y_{2}(v_{i})$ paths agree in
$U_{1}^{(n)}$.

This is very similar to the case of the first walk from Sections 4.3, 4.2 and
4.4, with only two differences : the random walk can stop inside $U_{1}$ and
the boundary of the two (slitted) domains are not exactly mapped to each other
by $\phi$ (note that in the statement, $\phi$ must be deterministic it cannot
be adapted for the slit domains we have at each step).

The first difference causes no issue whatsoever. It is now possible for the
sequence of time $T_{i}$ from the definition of the mixed loop-erasure 4.2 to
end at an odd step if a random walk hits one of the previous paths $Y(v_{i})$
before the boundary. This does not break the coupling since over these time
intervals $\tilde{X}^{1}=\tilde{X}^{2}$ and by induction the paths $Y(v_{i})$
agree in $U_{1}^{\prime}$.

The second difference could be more problematic. For example if the new
spanning tree branch $Y^{2}(v_{n+1})$ comes close to $Y^{2}(v_{1})$ without
hitting it, it could happen that $\phi(Y^{2}(v_{n+1}))$ intersects
$\tilde{Y}^{1}(v_{1})$. In this case a walk following $\phi(Y^{2}(v_{n+1}))$
might very well hit the boundary “too soon” and break the first part of the
proposition.

However this cannot happen if we take $\epsilon_{n}$ sufficiently small
compared to $\epsilon$. Indeed, the first time either $\tilde{X}_{1}$ or
$\tilde{X}_{2}$ come within $\epsilon_{n}$ of a boundary, the other walk must
be within $C\epsilon_{n}$ of a boundary and therefore both are very unlikely
to move more than $\epsilon$ from their current position and therefore cannot
erase any large loop. ∎

Similarly to the first case, once a large but finite number of paths have been
coupled in a neighbourhood of $U$, we can extend the coupling to the rest of
the tree by the finiteness theorem so overall we obtain Proposition 4.1. Using
Lemma 3.3 we obtain the second part of Theorem 1.1.

######  Proposition 4.11.

For any $\epsilon>0$, there exists $C>0$ such that for all $\delta$ small
enough

$\nu_{2}(\frac{1}{C}\leq\frac{d\nu_{1}}{d\nu_{2}}\leq C)\geq 1-\epsilon.$

######  Remark 4.12.

Now that we have upper and lower bound as in Propositions 3.4 and 4.11 in the
case where $D_{1}\subset D_{2}$, the result clearly extends to arbitrary
$D_{1}$ and $D_{2}$ by just introducing a intermediate common subset to both.
In fact in the above proof of Proposition 4.11 we did not really use the fact
that $D_{1}\subset D_{2}$ so we could have omitted the special case treated in
Section 3. However the fact that in the setting of Proposition 3.4 only a
simple conditioning is needed is worth noting and starting with the simpler
case is useful to the exposition. Also the proof of the simpler case will be
useful later for Corollary 5.1.

##  5 Extensions and generalisations

In this section, we provide small variants of our coupling in order to paint a
more global picture around Theorem 1.1.

###  5.1 Continuity with respect to the domains

As mentioned above, we did not keep track of the dependence on the domains of
the constants $C$ and $\epsilon$ from Theorem 1.1 but it is a very natural
question, the answer of which turns out to be a corollary of the proof of the
upper bound (Proposition 3.1). Note that this also covers the case of a non
smooth boundary in Theorem 1.1 and shows that the discretization setup used in
Section 2.1 could be replaced by any other reasonable one.

######  Corollary 5.1.

For any simply connected open sets $U,V$ with $\overline{U}\subset V$ and for
any $\epsilon>0$, there exists $r>0$ and $\delta_{0}$ such that for all simply
connected domains $D_{1},D_{2}$, if $V\subset D_{1}\cap D_{2}$ and the
Hausdorff distance between $\partial{D_{1}}$ and $\partial D_{2}$ is less than
$r$, then

$\nu_{1}(1-\epsilon\leq\frac{d\nu_{1}}{d\nu_{2}}\leq 1+\epsilon)\geq
1-\epsilon,\qquad\nu_{2}(1-\epsilon\leq\frac{d\nu_{1}}{d\nu_{2}}\leq
1+\epsilon)\geq 1-\epsilon.$

###### Proof.

In the proof of Proposition 3.1, note that the role of the event
$E^{\\#\delta}$ is precisely to reduce the problem to the case where $D_{1}$
and $D_{2}$ are very close. It is therefore easy to see that Lemma 3.2 applies
in this setting and that for $r$ small enough one has
$d_{TV}(\nu_{1}^{\\#\delta},\nu_{2}^{\\#\delta})\leq\epsilon^{2}$. We conclude
similarly as in Lemma 3.3 : by definition of the total variation distance we
known that $\int\lvert\frac{d\nu_{2}}{d\nu_{1}}-1\rvert d\nu_{1}\leq
2\epsilon^{2}$, and then we simply apply Markov’s inequality. ∎

In more abstract term, the corollary above says that the constant $C$ in
Theorem 1.1 (and even the whole marginal law $\nu$) depends continuously on
the domain for the topology defined by the Hausdorff distance on their
boundaries and that the continuity is even uniform if the boundary is far away
from $U$. The case of the continuity “close to the full plane” was actually
proved earlier.

######  Proposition 5.2.

For any bounded open set $U$ and $\epsilon>0$, there exists $R>0$ and
$\delta_{0}>0$ such that for all $\delta<\delta_{0}$, for all $D_{1},D_{2}$
which are either domains or the full plane, if $B(0,R)\subset D_{1}$ and
$B(0,R)\subset D_{2}$ then

$\nu_{1}(1-\epsilon\leq\frac{d\nu_{1}}{d\nu_{2}}\leq 1+\epsilon)\geq
1-\epsilon,\qquad\nu_{2}(1-\epsilon\leq\frac{d\nu_{1}}{d\nu_{2}}\leq
1+\epsilon)\geq 1-\epsilon.$

###### Proof.

The total variation distance between the two measures is controlled by Theorem
4.21 in [BLR19] if the aspect ratio between $U$ and the domains is large
enough. We conclude as in Corollary 5.1. ∎

###  5.2 Dimer configurations and height functions

As presented in Section 2.5, the uniform spanning tree is deeply connected to
the dimer model and its height function so it is natural to ask whether our
results can be generalised to that setting. In this section we will give a
positive answer to that question, with only a slight complication with respect
to the regularity assumption on the boundary.

As before consider $D_{1},D_{2},U$ open, simply connected such that
$\bar{U}\subset D_{1}\cap D_{2}$. In order to consider the dimer and height
function on them, further assume that $D_{1}$ and $D_{2}$ have a locally
connected boundary with a marked prime end (see Section 2.5 for details). We
will denote by $\nu_{1}^{(d)}$ (resp. $\nu_{2}^{(d)}$) the law of the
restriction of the dimer configurations in $D^{\\#\delta}_{1}$ (resp.
$D^{\\#\delta}_{2}$) to $U$. Similarly we denote by $\nu^{(h)}_{1}$ and
$\nu_{2}^{(h)}$ the laws of the restrictions of the height function to $h$.
Note that $\nu_{1}^{(d)}$ and $\nu_{1}^{(h)}$ are different because knowing a
dimer configuration on $U$ only determines the height function over $U$ up to
a global constant. We have a direct extension of Theorem 1.1

######  Theorem 5.3.

For any $\epsilon$, there exists $C>0$ such that for all $\delta$ small
enough,

$\nu_{1}^{(d)}(\frac{1}{C}\leq\frac{d\nu_{1}^{(d)}}{d\nu_{2}^{(d)}}\leq C)\geq
1-\epsilon,\quad\nu_{2}^{(d)}(\frac{1}{C}\leq\frac{d\nu_{1}^{(d)}}{d\nu_{2}^{(d)}}\leq
C)\geq 1-\epsilon,$

and similarly for the measures $\nu^{(h)}$.

###### Proof.

We first note that the statement is clearly stronger for the law on the height
function than for the law on dimers since the former determines the latter
locally. Also as noted in Remark 4.12, proving Proposition 3.1 is not really
necessary so let us focus on the proof of Proposition 4.1.

Compared to the initial case, we need to preserve the winding of UST branches
between $D_{1}$ and $D_{2}$. Recall (as pointed out in Remark 2.26) that with
our conventions, the value modulo $2\pi$ of the winding of a path only depends
on its initial direction. In particular for any choice of the map $\phi$ from
$D_{2}$ to $D_{1}$ and any face $f$ in $U$ we have
$W(\gamma_{f},\gamma_{f}(0))\equiv
W(\phi(\gamma_{f}),\phi(\gamma_{f}(0)))[2\pi]$. Furthermore, it is not hard to
see that if $\phi$ extends to a homeomorphism from $\overline{(D_{2})}_{d}$ to
$\overline{(D_{1})}_{d}$ and if it sends the marked prime end of $d_{2}$ to
the one of $D_{1}$, then one must have
$W(\gamma_{f},\gamma_{f}(0))-W(\phi(\gamma_{f}),\phi(\gamma_{f}(0)))=2k\pi$
for some $k\in\mathbb{Z}$ which does not depend on $f$. Finally it is easy to
see that composing if necessary with a map that twists $-k$ times the
topological annulus $D_{2}\setminus U$, we can always choose $\phi$ so that it
preserves the winding of paths starting in $U$. Since $X_{2}$ is absolutely
continuous with respect to a simple random walk, we can apply Lemma 2.5 to it.
In particular we can without loss of generality assume that $X_{2}(T_{\max})$
is at a macroscopic distance $r$ from the marked point of $\partial D_{2}$.

Now assume that in the construction of the walk $X_{1}$ (which is done exactly
as in Section 4.3), we choose rectangles small enough compared to $r$. In the
last step, by Beurling estimate, the diameter of the last added piece is very
unlikely to be greater than $r/4$ and in particular with high probability the
endpoint of $X_{1}$ must be within $d_{D_{1}}$-distance at most $r/2$ of
$\phi(X_{2}(T_{\max}))$. Overall since the winding depends continuously on the
path away from $U$, by taking $r$ small enough we can ensure that the windings
of the UST branches in $D_{1}$ and $D_{2}$ are at most $\pi/2$ away from each
other. On the other hand as pointed out in Remark 2.26, the value modulo
$2\pi$ of the winding is determined independently of the tree, so the only
possibility is that the windings of the UST branches in $D_{1}$ and $D_{2}$
agree. The rest of the proof follows as in Sections 4.4 and 4.5.

∎

The continuity statement is slightly more involved because, as already pointed
out in Section 2.4 the classical topology on domains are not strong enough for
us and we have to use the notion introduced in Definition 2.18.

######  Lemma 5.4.

For all $U$, $V$ simply connected domains with $\overline{U}\subset V$ and all
$\epsilon>0$, there exists $r>0$ such that for all $\delta$ small enough, for
all simply connected domains $D_{1},D_{2}$ with locally connected boundary and
marked prime end, if $D_{1}$ and $D_{2}$ are $r$ close then

$\nu_{1}^{(d)}(1-\epsilon\leq\frac{d\nu_{1}^{(d)}}{d\nu_{2}^{(d)}}\leq
1+\epsilon)\geq
1-\epsilon,\qquad\nu^{(d)}_{2}(1-\epsilon\leq\frac{d\nu^{(d)}_{1}}{d\nu^{(d)}_{2}}\leq
1+\epsilon)\geq 1-\epsilon,$

and similarly for $\nu^{(h)}$.

###### Proof.

As in Corollary 5.1, it is enough to show that with high probability we can
couple the laws of single branches in $D_{1}$ and $D_{2}$ so that they agree
up to a neighbourhood of the boundary and create the same winding.

Fix a face in $U^{\\#\delta}$ and let $X^{1}$, $X^{2}$, $\gamma_{1}$ and
$\gamma_{2}$ denote the random walks and tree branches (going to infinity
according to the convention from Section 2.5) in $D_{1}^{\\#\delta}$ and
$D_{2}^{\\#\delta}$ respectively. Fix $\epsilon>0$, $r$ to be chosen small
enough later and let $f:D_{1}\to D_{2}$ be an homomorphism given by Definition
2.18. We couple $X^{1}$ and $X^{2}$ by setting $X^{1}=X^{2}$ up to
$T:=\inf\\{t:d(X^{1}[t],\partial(D_{1}\cap D_{2})\leq 2r\\}$ and then letting
them move independently. Reasoning as in Corollary 5.1, it is easy to see that
if $r$ is small enough compared to $d(\partial U,\partial V)$ then
$\gamma_{1}$ and $\gamma_{2}$ agree on $U$ with high probability. We also let
$\tau_{1}$ and $\tau_{2}$ denote respectively the exit time of $D_{1}$ by
$X^{1}$ and $D_{2}$ by $X^{2}$. Finally let $x_{{\text{ref}}}$ denote the
marked prime end of $D_{1}$.

By Lemma 2.4 and Lemma 2.5, we know that there exists $\alpha>0$ such that

$\mathbb{P}(X^{1}[\tau_{1},T],X^{1}[\tau_{1}])\geq\sqrt{r})\leq
r^{\alpha},\quad\mathbb{P}(d_{D_{2}}(X^{2}[T],X^{2}[\tau_{2}])\geq\sqrt{r})\leq
r^{\alpha}$

and

$\mathbb{P}(\lvert X^{1}[\tau_{1}]-x_{{\text{ref}}}\rvert\leq 4\sqrt{r})\leq
r^{\alpha}\quad\mathbb{P}(\lvert X^{2}[\tau_{2}]-f(x_{{\text{ref}}})\rvert\leq
4\sqrt{r})\leq r^{\alpha},$

Putting the first two estimates together with the definition of $f$, we see
that with high probability the distance between $X^{2}[0,\tau_{2}]$ and
$f(X^{1}[\tau_{1}])$ is at most $3\sqrt{r}$. Furthermore, we see from the last
estimate that $X^{2}[\tau_{2}]$ and $f(X^{1}[\tau_{1}])$ cannot be on opposite
sides of $f(x_{{\text{ref}}})$ so the extensions along $\partial D_{1}$ and
$\partial D_{2}$ in the definition of $\gamma_{1}$ and $\gamma_{2}$ must also
be close to each other. By continuity of the winding,
$W(\gamma_{1},\gamma_{1}(0))$ and $W(\gamma_{2},\gamma_{2}(0))$ must therefore
be close (still with high probability) and by Remark 2.26 in that case they
must be equal. ∎

For restriction of large domains, the law of the global shift in the height
function depends on the whole domain so we only have a statement for
$\nu^{(d)}$. Also in that case the boundary regularity is essentially
irrelevant as long as there is some way to define a dimer configuration.

######  Lemma 5.5.

For any bounded open set $U$ and $\epsilon>0$, there exists $R>0$ and
$\delta_{0}>0$ such that for all $D_{1},D_{2}$ which are either domains or the
full plane, if $B(0,R)\subset D_{1}$ and $B(0,R)\subset D_{2}$ then

$\nu^{(d)}_{1}(1-\epsilon\leq\frac{d\nu^{(d)}_{1}}{d\nu^{(d)}_{2}}\leq
1+\epsilon)\geq
1-\epsilon,\qquad\nu^{(d)}_{2}(1-\epsilon\leq\frac{d\nu^{(d)}_{1}}{d\nu^{(d)}_{2}}\leq
1+\epsilon)\geq 1-\epsilon.$

###### Proof.

Fix $\epsilon>0$, it is well known that, for the full plane measure on trees,
there exits $R>0$ such that, with probability at least $1-\epsilon$, all
branches starting in $U$ merge within $B(0,R)$. In that case, the height
difference between any two points in $U$ is given by the winding of a path
staying inside $B(0,R)$ and in particular the dimer measure in $U$ is
measurable with respect to the tree measure in $B(0,R)$. We conclude by
applying Proposition 5.2 with $U$ replaced by $B(0,R)$. ∎

Another natural question is whether the global shift of the height function
can be chosen (somewhat) independently of the dimer configuration, especially
considering that the dimer configuration depends mostly on restriction of the
tree in $U$ while the height depend on the number of turns of branches around
$U$. The next two results give a positive answer to this question.

######  Lemma 5.6.

Let $D$ be a simply connected bounded domain with a locally connected boundary
and a marked boundary point and let $U$ be an open set such that
$\overline{U}\subset D$. For all $\ell\in\mathbb{Z}$ and $\epsilon$ there
exists $C>0$ such that for all $\delta$ small enough,

$\nu^{(h)}\big{(}\frac{1}{C}\leq\frac{d\nu^{(h)}(.+\ell)}{d\nu^{(h)}}\leq
C\big{)}\geq 1-\epsilon,$

where by $\frac{d\nu^{(h)}(.+\ell)}{d\nu^{(h)}}$ we denote the Radon-Nikodym
derivative of a height configuration shifted by $\ell$ with respect to the
original measure.

###### Proof.

This is essentially a corollary of the first part of the proof of Theorem 5.3
: in that proof we can take $D_{1}=D_{2}$ but consider a map $\phi$ that adds
exactly $2\pi\ell$ to the winding of any path from $U$ to $\partial D$ instead
of a map that preserves the winding. ∎

######  Corollary 5.7.

Let $D$ be a simply connected bounded domain with a locally connected boundary
and a marked boundary point and let $U$ be an open set such that
$\overline{U}\subset D$. For all $\epsilon>0$, there exists $C>0$ such that
for all $\delta$ small enough, there exists a set $\mathcal{E}$ of dimer
configurations of $U^{{\\#\delta}}$ such that

  * •

$\nu^{(d)}(\mathcal{E})\geq 1-\epsilon$,

  * •

$\forall\omega\in\mathcal{E},\nu^{(h)}\big{(}\forall\ell\in\mathbb{Z},|\ell|\leq\frac{1}{\epsilon},\frac{1}{C}\leq\frac{d\nu^{(h)}(.+\ell)}{d\nu^{(h)}}\leq
C\big{|}\omega\big{)}\geq 1-\epsilon$,

where in the second line, $\nu^{(h)}(\cdot|\omega)$ is the conditional law on
the height function in $U$ given that the dimer configuration in $U$ is given
by $\omega$.

###### Proof.

First by union bound, Lemma 5.6 implies that

$\nu^{(h)}\big{(}\forall\ell\in\mathbb{Z},|\ell|\leq\frac{1}{\sqrt{\epsilon}},\frac{1}{C}\leq\frac{d\nu^{(h)}(.+\ell)}{d\nu^{(h)}}\leq
C\big{)}\geq 1-\sqrt{\epsilon}.$

Let $G$ denote this event. We have
$\mathbb{E}_{\nu^{(d)}}(\nu^{h}(G^{c}|\omega))\leq\sqrt{\epsilon}$ so by
Markov inequality,

$\nu^{(d)}(\nu^{(h)}(G^{c}|\omega)\geq\epsilon^{1/4})\leq\epsilon^{1/4},$

which concludes the proof taking
$\mathcal{E}=\\{\omega:\nu^{(h)}(G|\omega)\geq 1-\epsilon^{1/4}\\}$. ∎

######  Remark 5.8.

Corollary 5.7 essentially shows that it is possible to sample the dimer
configuration in $U$ without revealing too much information over the global
shift of the height in the sense that it cannot be concentrated in intervals
smaller than $1/\epsilon$. A similar statement was obtained previously in
[Las18] when $U$ is small enough compared to $D$. The method there follows
more closely Wilson’s algorithm in order to identify concrete random variables
which are independent of the dimer configuration in $U$ but contribute to the
global shift of height function through a notion of “isolated scale”. This
could certainly be adapted to the setting of this paper if one needed a more
explicit construction of the set $\mathcal{E}$ or a more concrete link between
$\epsilon$, the constant $C$ and the domain $D$.

###  5.3 Sampling an annulus

In the previous sections, we always assumed $U$, $D_{1}$, $D_{2}$ to be simply
connected. While relaxing this assumption for $D_{1}$ or $D_{2}$ introduces
far to much complication for our purpose (see [BLR22] for information on
dimers and Temperley’s bijection in non simply-connected domains), it makes
sense to consider a general open set $U$. For example, for possible
applications to loop-ensembles it is very natural to ask whether one can
reveal the existence of a loop in a specific annulus without affecting too
much the distribution inside.

The first step is to prove that when running Wilson’s algorithm, it does not
“cost” to much to avoid a finite number of sets. More precisely, let $D$ be a
simply connected domain and let $U_{1}^{\prime},\ldots U_{n}^{\prime}$ be
disjoint open sets with $\overline{U_{i}^{\prime}}\subset D$ and let
$U_{1},\ldots,U_{n}$ be simply connected open sets with
$\overline{U_{i}}\subset U_{i}^{\prime}$. Define $V=D\setminus\cup
U^{\prime}_{i}$. We will denote by $\mathcal{T}^{ex}_{V}$ the union of all
branches starting in $V$ (seen as a tree) and call its law $\nu^{ex}_{V}$
while we call $\mathcal{T}_{V}$ and $\nu_{V}$ the restriction of the UST to
$V$ and its law.

######  Lemma 5.9.

For all $\epsilon>0$, there exists $p>0$ such that for all $\delta$ small
enough there exists a law $\tilde{\nu}_{V}$ such that

$\tilde{\nu}^{ex}_{V}(\mathcal{T}\cap(\cup U_{i})=\emptyset)\geq
1-\epsilon,\quad
d_{TV}(\tilde{\nu}_{V},\nu_{V})\leq\epsilon,\quad\frac{d\tilde{\nu}^{ex}_{V}}{d\nu^{ex}_{V}}\leq\frac{1}{p}.$

Furthermore the total variation bound also holds for the restriction of the
dimer height function and the dimer configuration.

###### Proof.

Fix $u_{1}\in U_{1},\ldots,u_{n}\in U_{n}$. By Proposition 4.11 from [BLR19],
a single loop-erased random walk started close to $V$ is unlikely to come very
close to any of the $u_{i}$. Combined with Schramm’s finiteness theorem, we
see that we can find $r>0$ such that $\nu^{ex}_{V}\mathcal{T}\cap(\cup
B(u_{i},r))=\emptyset)\geq 1-\epsilon$.

Take $\phi$ an homomorphism from $D$ to itself which restricts to the identity
over $D$ and such that $\phi(B(u_{i},r))=U_{i}$. We can construct
$\tilde{\nu}^{ex}_{V}$ as in Section 4, we just need to replace the last
condition in Section 4.3 (about the behaviour when the walk comes close to
$\partial D_{2}$) by the condition that the walk does not enter $\cup
B(u_{i},r)$. The proof for the spanning tree then follows exactly as above.

In this setting, the construction automatically preserves the winding of the
branches (and hence the height function and dimer configuration) because the
winding of a portion of path in some $U_{i}$ only depends on the points where
it enters and exits $U_{i}$. ∎

Now in each of the $U_{i}$, we can apply the previous results comparing the
law in the remaining region after sampling $\mathcal{T}^{ex}_{V}$ with the
original domain to get the following.

######  Proposition 5.10.

Let $\nu_{V,U_{i}}$ (resp. $\nu_{V,U_{i}}^{(h)}$, $\nu_{V,U_{i}}^{(d)}$) be
the law of the restriction of the tree (resp. the height function and dimer
configuration) to $V\cup\cup_{i}U_{i}$ and let $\nu_{V}$, $\nu_{U_{i}}$ be the
corresponding marginal distributions. For all $\epsilon>0$ there exists $C>0$
such that for all $\delta$ small enough

$\displaystyle\nu_{V,U_{i}}\big{(}\frac{1}{C}\leq\frac{d\nu_{V,U_{i}}}{d\nu_{V}\otimes\nu_{U_{1}}\otimes\ldots\otimes\nu_{U_{n}}}\leq
C\big{)}\geq 1-\epsilon,$
$\displaystyle\nu_{V}\otimes\nu_{U_{1}}\otimes\ldots\otimes\nu_{U_{n}}\big{(}\frac{1}{C}\leq\frac{d\nu_{V,U_{i}}}{d(\nu_{V}\otimes\nu_{U_{1}}\otimes\ldots\otimes\nu_{U_{n}})}\leq
C\big{)}\geq 1-\epsilon.$

The same holds for the dimer and height function measures.

In other words, the true joint law of the trace of the UST (or dimer/height
function) is comparable with an independent product of the marginal
distribution, in the Radon-Nikodym sense.

###### Proof.

We start by the equation in the first line of the display, where the lower
bound is the non-trivial part. Also we focus first on the laws on trees.

Let $\tilde{\nu}$ be defined as follow. First we sample $\mathcal{T}_{V}^{ex}$
as in Lemma 5.9, with the additional condition that $\mathcal{T}_{V}^{ex}$
does not come within distance $r$ of $\cup U_{i}$ for some $r>0$, which only
amounts to changing the definition of the $U_{i}$. Then in each $U_{i}$, we
sample independently the tree with boundary conditions given by
$\mathcal{T}_{V}^{ex}$, using in each case the law from Proposition 4.1 to
compare it with the law of $\nu_{U_{i}}$.

By construction $\tilde{\nu}$ satisfies
$d_{TV}(\tilde{\nu},\nu_{V}\otimes\nu_{1}\otimes\ldots\otimes\nu_{U_{n}})\leq\epsilon$.
Furthermore by the continuity result Corollary 5.1, the constants appearing in
each use of Proposition 4.1 can be chosen uniformly over all realizations of
$\mathcal{T}^{ex}_{V}$, therefore $\frac{d\tilde{\nu}}{d\nu_{V,U_{i}}}\leq
1/p$ for some $p$ since this is true for the marginal on
$\mathcal{T}_{V}^{ex}$ and then true for the conditional laws of all
restriction to the $U_{i}$ given $\mathcal{T}_{V}^{ex}$. Given this coupling,
we conclude as in the proof of Proposition 4.11. The proof of the second
equation is similar.

Finally for the laws on dimers or height function, the proof is very similar
except that the weaker continuity result from Lemma 5.4 has to be used instead
of Corollary 5.1. More precisely fix two intermediate open set
$V^{\prime},V^{\prime\prime}$ such that $\overline{V}\subset V^{\prime}$,
$\overline{V^{\prime}}\subset V^{\prime\prime}$ but
$\overline{V}^{\prime\prime}\cap\overline{U}_{i}=\emptyset$ for all $i$. By
the finiteness theorem Lemma 2.14, choose $k$ points in $V^{\prime\prime}$
such that, given the subtree $\mathcal{T}_{k}$ emanating from these points,
except on a event of probability $\epsilon$ all random walks and all branches
started in $V$ will stay in $\partial V^{\prime\prime}$ and no branch started
from $D\setminus V^{\prime\prime}$ can enter $V^{\prime}$. It is well known
(see for example Section 3.4 in [LSW04]) that the laws of loop-erased random
walk are tight (as $\delta\to 0$) when viewed as simple curves for the uniform
topology on curve up to reparametrisation. Therefore, using the fact that only
$k$ branches have been sampled, by Lemmas 5.4 and 2.20, for all $\epsilon$ we
can find $C$ such that with high probability Theorem 5.3 can be used with the
constant $C$ to compare the $\nu_{U_{i}}(\cdot|\mathcal{T}_{k})$ to the
$\nu_{u_{i}}$. This concludes because, by construction, the restrictions to
the $U_{i}$ and $\mathcal{T}_{V}$ are almost independent given
$\mathcal{T}_{k}$. ∎

###  5.4 Continuum limit

In this section, we will show the analogue of Theorem 1.1 for the continuum
spanning tree.

######  Proposition 5.11.

Let $D_{1}$, $D_{2}$ be two simply connected domains and let $U$ be an open
set such that $\overline{U}\subset D_{1}\cap D_{2}$. Let $\nu_{1}^{c}$ and
$\nu_{2}^{c}$ be the laws of the restriction to $U$ of the continuum spanning
tree in $D_{1}$ and $D_{2}$. These laws are absolutely continuous with respect
to each other.

Before giving the proof which will follow from Theorem 1.1, let us note that
this results should be fairly easy to prove using the theory of imaginary
geometry. Indeed the restriction of the trees to $U$ are measurable with
respect to the restriction of the associated GFFs to $U$ but since the GFF is
a Gaussian process, it should be possible to write the Radon-Nicodym
derivative explicitly. However we still think that the following proof is
useful because it does not rely on any knowledge of SLE or the GFF. For
example it sounds much easier to adapt this proof to a non simply-connected
setting than to develop imaginary geometry on Riemann surfaces.

###### Proof.

In this proof, consider $G^{\\#\delta}=\delta\mathbb{Z}^{2}$ with unit weight.
By the celebrated result of [LSW04], the laws $\mu_{i}^{\\#\delta}$ of the
discrete uniform spanning tree in $\mathbb{D}_{i}^{\\#\delta}$ converge in law
to continuum limits $\mu_{i}^{c}$ say for Schramm’s topology which defines a
Polish space.

Fix $\epsilon>0$ and write

$\bar{\nu}_{i}^{\\#\delta}=\nu_{i}^{\\#\delta}1_{\frac{1}{C}\leq\frac{d\nu_{1}^{\\#\delta}}{d\nu_{2}^{\\#\delta}}\leq
C},$

where $C$ is obtained from Theorem 1.1 with $i=1,2$. Note that, since
$\bar{\nu}_{i}^{\\#\delta}\leq\nu_{i}^{\\#\delta}$ and the sequence
$\nu_{i}^{\\#\delta}$ converge in law, the sequence
$\bar{\nu}_{i}^{\\#\delta}$ must be tight so let $\bar{\nu}_{i}^{c}$ be a sub-
sequential limits. By the choice of $C$ in Theorem 1.1, both
$\bar{\nu}_{1}^{c}$ and $\bar{\nu}_{2}^{c}$ have total mass at least
$1-\epsilon$.

By definition, for any positive continuous bounded function $f$,
$\bar{\nu}_{1}^{\\#\delta}(f)\leq\nu_{1}^{\\#\delta}(f)$ so taking the limit
along the appropriate subsequence $\bar{\nu}_{1}(f)^{c}\leq\nu_{1}^{c}(f)$ and
therefore $\nu_{1}^{c}-\bar{\nu}_{1}^{c}$ is a positive bounded linear form on
continuous functions. By the Riesz representation theorem, it extends to a
unique positive measure but since $\nu_{1}^{c}-\bar{\nu}_{1}^{c}$ is already a
measure we must have $\bar{\nu}_{1}(E)^{c}\leq\nu_{1}^{c}(E)$ for any
measurable event $E$. Of course the analogous statement holds for
$\bar{\nu}^{c}_{2}$ and $\nu^{c}_{2}$. Now note that the same proof and the
definition of the $\bar{\nu}_{i}^{c}$ also shows that

$\frac{1}{C}\bar{\nu}_{2}^{c}(E)\leq\bar{\nu}^{c}_{1}(E)\leq
C\bar{\nu}_{2}^{c}(E).$

Combining the above with the bound on the total mass of the $\bar{\nu}_{i}$,
we obtain that

$\frac{1}{C}(\nu_{2}^{c}(E)-\epsilon)\leq\nu^{c}_{1}(E)\leq C\
\nu_{2}^{c}(E)+\epsilon$

for any event $E$ and this concludes the proof since $\epsilon$ was arbitrary.
∎

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