3.1 The height function

It is well known that $3$ -colorings of $\mathbb {Z}^{2}$ can be represented as homomorphisms from $\mathbb {Z}^{2}$ to $\mathbb {Z}$ . Recall that a homomorphism $\varphi $ from $D \subseteq \mathbb {Z}^{2}$ to $\mathbb {Z}$ is a map such that $|\varphi (x)-\varphi (y)|=1$ whenever $x,y \in D$ are adjacent. We also refer to such homomorphisms as height functions. We always work with height functions which are even on the even sublattice of $\mathbb {Z}^{2}$ .

Suppose that D is simply connected. It can be checked that the mapping $h \mapsto h\, \mod 3 $ is a bijection from the space of height functions whose value is fixed on some vertex of D to the space of 3-colorings of D whose color is fixed on that same vertex. The inverse mapping can be defined as follows. If $\mathsf c$ is a $3$ -coloring, then the gradient of h along the directed edge $e=(u,v)$ is given by $\nabla h (e) =+1$ if $\mathsf c(v) - \mathsf c(u) \in \{1,-2\}$ and $-1$ if $\mathsf c(v) - \mathsf c(u) \in \{-1,2\}$ . In particular, given a $3$ -coloring of $\mathbb {Z}^{2}$ , this mapping defines the gradient of a height function h on $\mathbb {Z}^{2}$ . In general, height functions will come with a predefined set of boundary conditions (a precise definition will follow) which will always determine the function from its gradient.

Suppose that D is finite and simply connected. Let $\phi _{D}^{01}$ denote the uniform measure on height functions on D whose values on the boundary $\partial D$ are fixed to be $0$ and 1 on even and odd vertices, respectively. Then $\phi _{D}^{01}$ is pushed forward to $\mu ^{01}_{D}$ by the modulo 3 map.

3.2 Preliminary results

We begin with some notation. For any m and n, denote $\Lambda _{m,n} = [-m,m] \times [-n,n] \cap \mathbb {Z}^{2}$ , and recall that $\Lambda _{n}=\Lambda _{n,n}$ . For $n \ge m$ , let $A_{m,n}$ denote the annulus $\Lambda _{n} \setminus \Lambda _{m}$ . A path in $\mathbb {Z}^{2}$ is a sequence of vertices $(v_{1},v_{2},\ldots , v_{n})$ such that $v_{i}$ is adjacent to $v_{i+1}$ for $i=1,\ldots , n-1$ . A path is a loop if $v_{i} \neq v_{j}$ for all $1\le i \neq j \le n-1$ and $v_{1} = v_{n}$ . We also require a notion of diagonal connectivity in $\mathbb {Z}^{2}$ : say that u is a $\times $ -neighbor of v if u and v are at Euclidean distance $\sqrt {2}$ (that is, they are diagonal neighbors). We define a $\times $ -path ( $\times $ -loop) in a similar way, replacing the $\mathbb {Z}^{2}$ adjacency by $\times $ -adjacency. Note that a $\times $ -path consists of vertices of a single parity, so that we may talk about even and odd $\times $ -paths. A domain is a simply connected finite subset D of $\mathbb {Z}^{2}$ whose boundary $\partial D$ is entirely contained in the even lattice. When D is a domain, we write $\mu ^{0}_{D}$ and $\phi ^{0}_{D}$ for $\mu ^{01}_{D}$ and $\phi ^{01}_{D}$ , respectively.

A key tool we need is the recently established Russo–Seymour–Welsh estimate for height functions [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9] (a classical estimate in planar percolation-type models), which was used in [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9] to show logarithmic variance for the height function. Let $\mathcal {H}_{h\ge k}(\Lambda _{\rho n,n})$ (respectively, $\mathcal {H}_{h=k}^{\times }(\Lambda _{\rho n,n})$ ) be the event that there is a path (respectively, $\times $ -path) with height at least k (respectively, equal to k) joining the left and right boundaries of $\Lambda _{\rho n,n}$ and lying completely inside $\Lambda _{\rho n,n}$ .

Theorem 3.2. (Russo–Seymour–Welsh estimate [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9])

For every $\varepsilon ,R,\rho ,k>0$ , there exists $c=c(\varepsilon ,R,\rho ,k)>0$ such that for any $n \ge {10k}/{\varepsilon \wedge \rho }$ and any domain $\Lambda _{\rho n,n} \subset D\subset \Lambda _{Rn}$ such that the distance between $\Lambda _{\rho n,n}$ and $\partial D$ is at least $\varepsilon n$ ,

(3.1) $$ \begin{align} c \le\ &\phi_{D}^{0} [\mathcal{H}_{h\ge k}(\Lambda_{\rho n,n})] \le 1- c, \end{align} $$

(3.2) $$ \begin{align} c \le\ &\phi_{D}^{0} [\mathcal{H}_{h=k}^{\times}(\Lambda_{\rho n,n})] \le 1-c. \end{align} $$

There are two key tools used in proving the above theorem, versions of which were also classically used to study planar percolation and random-cluster models [Reference Duminil-Copin8]. The first one is duality of paths, which roughly states that a left-to-right crossing of height $h \ge m$ is blocked by a top-to-bottom crossing of height $h<m$ . However, one needs to be careful with the type of connectivity used, and we will carefully point this out at the relevant place in the proof (rather than stating a general duality lemma, such as [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9, Lemma 2.4], which we do not need).

The second key tool is a monotonicity property for the height function, classically known as the Fortuin–Kasteleyn–Ginibre (FKG) inequality and lattice condition. To properly state this, we introduce a general notion of boundary condition. Given a domain D, a boundary condition is a pair $(B,\kappa )$ with $B \subset D$ and $\kappa $ a function that assigns a subset $\kappa _{v} \subset \mathbb {Z}$ to each $v \in B$ . Let $\text {Hom}( D,B,\kappa )$ denote the set of homomorphisms h on D such that $h_{v}\in \kappa _{v}$ for every $v\in B$ . We say that the boundary condition $(B,\kappa )$ is admissible if $\text {Hom}( D,B,\kappa )$ is non-empty and finite. For an admissible boundary condition $(B,\kappa )$ , we let $\phi ^{B, \kappa }_{D}$ denote the uniform measure on $\text {Hom}(D,B,\kappa )$ . When $B=\partial D$ , we omit B from the notation. A function $F\colon \mathbb {Z}^{D} \mapsto \mathbb {R}$ is increasing if $F(h) \ge F(h^{\prime })$ for any $h, h^{\prime } \in \mathbb {Z}^{D}$ satisfying $h_{v} \ge h^{\prime }_{v}$ for all $v \in D$ .

The first property is called the comparison between boundary conditions and the second the FKG inequality. We also crucially use monotonicity properties of $|h|$ in addition to those of h. We say that the boundary condition $(B,\kappa )$ is $|h|$ -adapted if there exists a partition $B_{\text {pos}}(\kappa ) \sqcup B_{\text {sym}}(\kappa )$ of B such that:

• • for any $v \in B_{\text {pos}}(\kappa )$ , $\kappa _{v}\subset \mathbb {Z}_{+}:=\{1,2,\ldots \}$ ;

• • for any $w \in B_{\text {sym}}(\kappa )$ , $\kappa _{w} =-\kappa _{w}$ .

For example, if $w \in B_{\text {sym}}(\kappa )$ , $\kappa _{w}$ could be $\{-3,-2,2,3\}$ .

It is these monotonicity properties that make working with the height function representation beneficial for us.

3.3 Proof strategy

Given a height function h, a level loop is a $\times $ -loop on which the height is constant. We claim that the Russo–Seymour–Welsh estimate guarantees that under $\phi ^{0}_{D}$ , for a domain D containing $\Lambda _{n}$ but not containing a much larger box, most points in $\Lambda _{n}$ are surrounded by a level loop of height 0 contained entirely in $\Lambda _{n}$ . Indeed, if we look at exponential scales starting from the boundary inwards, there is a good chance of finding many successive level loops with height increment $\pm 2$ . Using the symmetry of the height function, we can argue that the actual increment is $+2$ or $-2$ with equal chance, and hence the heights along these nested loops constitute a simple random walk. Since a simple random walk is recurrent, there is a high chance of hitting a loop of height zero, thereby yielding the claim.

Now suppose we have two measures on height functions on D, one with boundary condition $\xi $ and one with boundary condition 0 (here we think of $\xi $ as specifying only the absolute values on the boundary). By the FKG inequality for $|h|$ , we can couple samples $h^{\xi }$ and $h^{0}$ from these measures so that inside the outermost level loops of $0$ for $h^{\xi }$ , the two height functions agree. Indeed, the FKG inequality for $|h|$ implies that $|h^{\xi }|$ stochastically dominates $|h^{0}|$ , which means that we can couple $h^{\xi }$ and $h^{0}$ so that $|h^{\xi }|$ dominates $|h^{0}|$ pointwise. However, since 0 is the lowest absolute value, the outermost level loop of height 0 for $h^{\xi }$ must also be a level loop of height 0 for $h^{0}$ , and hence, by the domain Markov property, we can couple them inside to be the same. There are some technical issues with applying this argument as is, but these can be handled using standard exploration procedures.

However, combining the above two arguments is not enough, since when the boundary condition $\xi $ is very high in absolute value, there is a good chance that there is no outermost level loop of height 0, and indeed the height function delocalizes as $\Lambda _{n}$ becomes large [Reference Chandgotia, Peled, Sheffield and Tassy4, Reference den Hollander and Steif9]. Thus, we need to reduce to the case where the boundary condition $\xi $ can be taken to be not too large. For this, we show that with high probability in the Gibbs measure of the coloring, we can find a loop $\mathcal {L}$ of color 0 in the annulus $A_{n,\Delta n}$ for some large $\Delta $ . Now, by the domain Markov property, the law of the coloring inside the domain enclosed by $\mathcal {L}$ can be viewed as a uniform homomorphism height function with boundary conditions 0 on $\mathcal {L}$ . Therefore, with high probability, we have somewhat reduced to the desired case above (the boundary condition is on the loop rather than on the boundary of the box). Note that we need to switch back to the height function representation of the coloring to establish the  coupling since we are crucially using the FKG inequality for the absolute value of the height function to that end.

By employing a two-step iteration of the above argument, one may establish the required coupling needed to verify the local mixing condition. However, this would not yield a power-law rate function. The reason for this stems from the fact that, with probability that is polynomial in k, a simple random walk does not return to 0 within k steps, whereas the number of steps k is with high probability only logarithmic in n, so that the former argument would yield a rate function $\rho $ decaying as $\rho (n) \le C(\log n)^{-\alpha }$ . To obtain a power-law rate, we apply the above argument iteratively, switching back and forth between the coloring and height function representations. Such a back-and-forth procedure seems essential for this (see Corollary 4.6). Roughly speaking, we first find the outermost monochromatic loop in the coloring. We then we switch to the height function and start looking for a level loop of height 0, allowing ourselves to stop if we do not find such a loop after searching a constant number of scales. If we find such a loop quickly, then we are done. Otherwise, we give up on finding the level loop of height 0 in this iteration, and proceed to the next iteration by going back to the coloring in order to find the next outermost monochromatic loop. Repeating this procedure yields a coupling which establishes local mixing with a power-law rate.

4.1 Level lines of the height function

In this section we prove some results about the level loops of the height function. Recall that a level loop of a height function is a $\times $ -loop on which the height is constant. We will mostly be focused on level loops in the even lattice, or equivalently, level loops on which the height is even.

Let D be a domain containing the origin and let h be a height function on D sampled from $\phi ^{0}_{D}$ . We inductively define a sequence of nested level loops $\mathcal {L}_{0},\ldots ,\mathcal {L}_{M}$ surrounding the origin as follows. First, we set $\mathcal {L}_{0}$ to be the boundary loop of the domain D. Next, we let $\mathcal {L}_{1}$ be the outermost level loop with $|h| = 2$ surrounding the origin. Now suppose that we have already defined the level loop $\mathcal {L}_{m}$ and that the height on it is $H_{m}\in 2\mathbb {Z}$ . Let $\mathcal {L}_{m+1}$ be the outermost level loop surrounding the origin in the domain enclosed by $\mathcal {L}_{m}$ and having height $H_{m+1}$ satisfying $|H_{m+1}-H_{m}| = 2$ . If no such loop exists, then we set $M=m$ and stop the inductive procedure. We say that the level loops $\mathcal {L}_{0},\ldots ,\mathcal {L}_{M}$ are essential, and note that between $\mathcal {L}_{m}$ and $\mathcal {L}_{m+1}$ there may be many non-essential level loops of height $H_{m}$ surrounding the origin.

Observe that, conditioned on the exploration up to $\mathcal {L}_{m}$ , the height difference $H_{m+1}-H_{m}$ is uniform in $\{-2,2\}$ . Indeed, given the exploration, the map $h \mapsto 2H_{m} - h$ inside the domain enclosed by $\mathcal {L}_{m}$ is an involution which inverts the sign of $H_{m+1}-H_{m}$ . Therefore, conditioned on the entire sequence of essential level loops $\mathcal {L}_{0},\ldots ,\mathcal {L}_{M}$ , the random variables $\{H_{m+1}-H_{m}\}_{m=0}^{M-1}$ are i.i.d. uniform $\pm 2$ . In particular, conditioned on M, the sequence $(\tfrac 12 H_{m})_{m=0}^{M}$ defines an M-step simple symmetric random walk starting from 0.

Our main technical result about the level lines of the height function is the following, which shows that, with high probability, the number of essential level loops in an annulus is linear in the logarithm of the aspect ratio of the outer and inner boundaries of the annulus. More specifically, we provide an upper bound on the number of such loops that intersect the annulus and a lower bound on the number of such loops that are entirely contained in the annulus.

Proposition 4.1. There exist constants $C,c>0$ such that the following assertion holds. Let $k \ge 200$ , let $a \ge 1$ and let D be a domain. Let $N=N(k,a)$ be the number of essential level loops contained in $A_{k,2^{a}k}$ and let $N^{\prime } = N^{\prime }(k,a) \ge N$ be the number of essential level loops intersecting $A_{k,2^{a}k}$ . Then

(4.1) $$ \begin{align} \phi_{D}^{0}(N^{\prime} \ge n) \le e^{-cn \log (\frac na \wedge k)} \quad\text{for any }n \ge Ca, \end{align} $$

and if D contains $\Lambda _{2^{a}k}$ then

(4.2) $$ \begin{align} \phi_{D}^{0}(N \le ca) \le Ce^{-ca}. \end{align} $$

We point out that there is no assumption on the domain in (4.1).

4.1.1 Corollaries to Proposition 4.1

We give four corollaries to Proposition 4.1 here. The first two will be used for the proof of Theorem 3.1, whereas the second two will not and simply provide additional information. The former two focus on level loops whose heights are multiples of 6. Such level loops correspond in the coloring representation to monochromatic loops of color 0 in the even lattice. This conforms with the convention that the height function is even on the even lattice and with the fact that the coloring is the modulo 3 of the height function.

The first corollary shows that the origin is typically surrounded by a level loop of height 0 modulo 6 which lies inside an annulus of a fixed aspect ratio. This will later allow us to relate various boundary conditions which are arbitrarily far away from a given box to zero boundary conditions which are near the boundary of the box.

Corollary 4.2. There exist $C,\alpha>0$ such that the following assertion holds. Let $k \ge 1$ , let $\Delta>1$ and let D be a domain containing $\Lambda _{\Delta k}$ . Then

$$ \begin{align*} \phi^{0}_{D}(\Lambda_{k}\text{ is surrounded by a level loop of height }0\text{ modulo }6\text{ inside }\Lambda_{\Delta k}) \ge 1-C\Delta^{-\alpha}. \end{align*} $$

Moreover, the same bound holds under the measure $\phi ^{\kappa }_{D}$ for any finite simply connected set D containing $\Lambda _{\Delta k}$ and any admissible boundary condition $\kappa $ on $\partial D$ such that $\bigcup _{v \in \partial D} \kappa _{v}$ is contained in some interval of size $10$ .

Proof. Recall that, given M, the sequence $(\tfrac 12 H_{m})_{m=0}^{M}$ defines an M-step simple symmetric random walk. Let N be the number of essential level loops contained in $A_{k,\Delta k}$ . Since an n-step simple symmetric random walk (starting from anywhere) visits $3\mathbb {Z}$ with probability at least $1-Ce^{-cn}$ , it suffices to show that $\phi ^{0}_{D}(N \le c\log \Delta ) \le \Delta ^{-\alpha }$ . This in turn follows from (4.1).

Consider now the general case and suppose that $\kappa _{v} \subset [a+1,b-1]$ for all v and some $a,b \in 2\mathbb {Z}$ having $b-a=20$ . Let $\mathcal {D}$ be the largest domain (with respect to inclusion) contained in D and containing $\Lambda _{k}$ . It follows that every $v \in \partial \mathcal {D}$ is at distance at most $1$ from $\partial D$ . In particular, under $\phi ^{\kappa }_{D}$ , the height on $\partial \mathcal {D}$ must be between a and b everywhere. Thus, by the FKG inequality, we can couple $h^{-} \sim \phi ^{a}_{\mathcal {D}}$ , $h^{+} \sim \phi ^{b}_{\mathcal {D}}$ and $h \sim \phi ^{\kappa }_{D}$ so that $h^{-} \le h \le h^{+}$ and $h^{+}-h^{-}=20$ in $\mathcal {D}$ . (The existence of such a three-way coupling is not immediate from the stochastic ordering of $h^{-}$ , h, $h^{+}$ and the distribution equality $h^{+}=h^{-}+20$ , but follows easily using a Markov chain argument as in the proof of [Reference Grimmett13, Theorem 2.1].) In particular, the essential level loops $\mathcal {L}_{1},\ldots ,\mathcal {L}_{M}$ of $h^{-}$ and $h^{+}$ coincide. As before, the number N of essential level loops in $A_{k,\Delta k}$ is at least $c\log \Delta $ with probability at least $1-\Delta ^{-\alpha }$ . Note that if two of these N loops, say $\mathcal {L}_{i}$ and $\mathcal {L}_{j}$ , have height difference at least 26 in $h^{-}$ (equivalently, in $h^{+}$ ), then h must have a level loop of height 0 mod 6 somewhere in between $\mathcal {L}_{i}$ and $\mathcal {L}_{j}$ . Since an n-step simple random walk starting from some x reaches $x\pm 26$ with probability at least $1-Ce^{-cn}$ , the required bound follows.

The next corollary roughly says that if a domain contains a box whose size is of the same order as the in-radius of the domain, then, with constant probability, the first essential loop of height 0 modulo 6 inside the box will have height exactly 0 (not merely 0 modulo 6).

Corollary 4.3. There exist $K,c>0$ such that the following assertion holds. Let $k \ge K$ , let $\Delta \ge 2$ and let D be a domain such that $\Lambda _{k} \subset D \not \supset \Lambda _{\Delta k}$ . Then

$$ \begin{align*} \phi^{0}_{D}(\text{first essential level loop of height }0\text{ mod }6\text{ inside }\Lambda_{k}\text{ has height }0) \ge \frac c{\sqrt{\log\Delta}}. \end{align*} $$

Proof. Let N be the number of essential level loops contained in $\Lambda _{k}$ . Note that the first $N^{\prime }:=M-N$ loops in $\mathcal {L}_{1},\ldots ,\mathcal {L}_{M}$ are not contained in $\Lambda _{k}$ and therefore intersect the annulus $A_{k,\Delta k}$ . Given N and $N^{\prime }$ , and on the event that $N \ge 2$ , since $(\tfrac 12 H_{m})_{m=0}^{M}$ is a simple symmetric random walk starting from 0, the probability that the first essential level loop of height 0 mod 6 inside $\Lambda _{k}$ exists and has height 0 is at least the probability that a simple random walk of length $N^{\prime \prime } := N^{\prime }+1+{\textbf {1}}_{\{N^{\prime }\text { even}\}}$ ends at 0. The latter is $2^{-N^{\prime \prime }} \binom {N^{\prime \prime }}{N^{\prime \prime }/2} \ge c/\sqrt {N^{\prime \prime }}$ . Finally, Proposition 4.1 implies that $\phi ^{0}_{D}(N \ge 2,N^{\prime } \le C\log \Delta ) \ge \frac {1}{2}$ , as long as k is large enough. Combining these estimates yields the corollary.

The next corollary shows that small oscillations in the boundary conditions do not have much effect on the distribution of the height function deep inside the domain. This implies that the convergence of $\mu ^{ij}_{D}$ to $\mu $ extends to a wider class of boundary conditions; see Remark 4.5.

Corollary 4.4. For any $\varepsilon>0$ , there exists $\delta>0$ such that the following assertion holds. Let $n \ge k \ge 1$ , let D be a domain containing $\Lambda _{n}$ and let $\tau \in \mathbb {Z}^{\partial D}$ be an admissible boundary condition such that $|\tau _{v}| \le \delta \sqrt {\log ( n/k)}$ for all $v \in \partial D$ . Then

$$ \begin{align*} \operatorname{\mathrm{dist}}_{\mathrm{TV}}((\phi^{\tau}_{D})_{|\Lambda_{k}}, (\phi^{0}_{D})_{|\Lambda_{k}}) \le \varepsilon. \end{align*} $$

Proof. Denote $a := \lceil \log (n/k) \rceil $ and $b := \max |\tau _{v}| \le \delta \sqrt {a}$ . It suffices to show that the total variation distance between the restrictions of $\phi ^{b}_{D}$ and $\phi ^{-b}_{D}$ to $\Lambda _{k}$ is at most $\varepsilon $ , as the two measures $\phi ^{\tau }_{D}$ and $\phi ^{0}_{D}$ are stochastically between these by the FKG inequality. Let $h^{\pm }$ be sampled from $\phi ^{\pm b}_{D}$ , and coupled as follows. First, couple their essential level loops $\mathcal {L}_{1},\ldots ,\mathcal {L}_{M}$ to coincide. Second, couple their heights on these loops, denoted by $H^{\pm }_{m}$ , so that their absolute values increase/decrease together (that is, $H^{+}_{m+1}-H^{+}_{m} = -(H^{-}_{m+1}-H^{-}_{m})$ ) until the first time $M^{\prime }$ where they both reach $H^{\pm }_{M^{\prime }}=0$ , at which point one may couple $h^{+}$ and $h^{-}$ so that they coincide in the domain $\mathcal {D}$ enclosed by $\mathcal {L}_{M^{\prime }}$ . This coupling shows that $\operatorname {\mathrm {dist}}_{\mathrm {TV}}(h^{+}_{|\Lambda _{k}},h^{-}_{|\Lambda _{k}}) \le \mathbb {P}(\mathcal {D}\text { does not contain }\Lambda _{k})$ . Let N be the number of essential level loops surrounding $\Lambda _{k}$ . By (4.2), there exist a universal constant $c>0$ and a constant $a_{0}=a_{0}(\varepsilon )$ such that $\phi _{D}^{0}(N \ge ca) \ge 1-\varepsilon $ if $a \ge a_{0}$ . Standard estimates yield that a simple random walk of length $ca$ starting from $\lfloor \delta \sqrt {a} \rfloor $ hits 0 with probability at least $1-\varepsilon $ if $\delta =\delta (\varepsilon )>0$ is chosen small enough. Thus, $\phi _{D}^{0}(\Lambda _{k} \subset \mathcal {D}) \ge 1-2\varepsilon $ as long as $a \ge a_{0}$ . Finally, note that by decreasing $\delta $ if necessary, we can ensure that $a<a_{0}$ implies that $b=0$ , in which case the desired statement is trivial.

Remark 4.5. Recall that Theorem 3.1 implies that $\mu ^{ij}_{D}$ converges to $\mu $ as D increases to $\mathbb {Z}^{2}$ along simply connected finite subsets. Suppose that a feasible boundary condition $\xi \in \{0,1,2\}^{\partial D}$ has oscillation at most m if there is a height function $\tau $ on D such that $\xi $ equals $\tau $ modulo 3 on $\partial D$ and $|\tau |\le m$ on $\partial D$ . Then Corollary 4.4 implies that $\mu ^{\xi }_{D}$ converges to $\mu $ as long as $\xi $ has oscillation at most $o(\sqrt {\log (r_{D})})$ , where $r_{D}$ is the largest n such that $\Lambda _{n} \subset D$ .

We end this section with a result about correlation decay for the height function and for the colorings. While the latter decays as a power law in the in-radius of the domain, the former only decays as a power law in the logarithm of the in-radius. This shows that a n rate in Theorem 3.1 is best possible. The argument for the colorings (corresponding to the statement about the height modulo 3) was suggested to us by Ron Peled.

Corollary 4.6. There exist constants $C,c,\alpha ,\beta>0$ such that for any $n \ge 2$ and any domain D having $\Lambda _{n} \subset D \not \supset \Lambda _{n+1}$ ,

$$ \begin{align*} \frac13 + \frac{c}{n^{\alpha}} \le \phi^{0}_{D}(\text{height at origin is }0\text{ modulo }3) \le \frac13 + \frac{C}{n^{\beta}} \end{align*} $$

and

$$ \begin{align*} \frac{c}{(\log n)^{3/2}} \le \phi^{0}_{D}(\text{height at origin is }0) - \phi^{2}_{D}(\text{height at origin is }0) \le \frac{C}{(\log n)^{3/2}}. \end{align*} $$

Proof. Let h be sampled from $\phi ^{0}_{D}$ and denote $X := h(0,0)$ . Since X is symmetric,

$$ \begin{align*} \mathbb{P}(X \in 3\mathbb{Z}) - \frac13 = \frac23\bigg[\mathbb{P}(X \in 3\mathbb{Z}) - \frac12 \mathbb{P}(X \notin 3\mathbb{Z})\bigg] = \frac23 \mathbb{E}\bigg[\cos\bigg(\frac{2\pi X}3\bigg)\bigg].\end{align*} $$

Recall that the essential level loops $\mathcal {L}_{1},\ldots ,\mathcal {L}_{M}$ were defined as the outermost level loops with $\pm 2$ height increments. Define level loops $\mathcal {L}^{\prime }_{1},\ldots ,\mathcal {L}^{\prime }_{N}$ similarly as the outermost level loops with $\pm 1$ increments (so that the essential level loops are a subset of these). Note that, given N, X is distributed as the sum of N independent uniform $\pm 1$ variables. Thus,

$$ \begin{align*} \mathbb{E}[\cos(\theta X) \mid N] = \Re \mathbb{E}[e^{i\theta X} \mid N] = \cos(\theta)^{N}. \end{align*} $$

As N is even (since X is), plugging in $\theta = {2\pi }/3$ , we get that $\mathbb {E}[\cos ({2\pi X}/3) \mid N] = 2^{-N}$ . Thus,

$$ \begin{align*} \mathbb{P}(X \in 3\mathbb{Z}) = \frac13 + \frac23 \mathbb{E}[2^{-N}]. \end{align*} $$

It remains to bound $\mathbb {E}[2^{-N}]$ from above and below. By tuning the constants in the statement, we may assume that n is sufficiently large. To obtain the stated upper bound, it suffices to show that $\mathbb {P}(N \le c\log n) \le n^{-\zeta }$ for some universal constants $c,\zeta>0$ . Since $N \ge M$ , this follows from (4.2). For the lower bound, it suffices to show that $\mathbb {P}(N \le C\log n) \ge \tfrac 14$ for some universal constant $C>0$ . It follows from (4.1) that $\mathbb {P}(M \le C\log n) \ge \tfrac 12$ . To obtain the bound for N, note that, given N, M is distributed as $\max \{m : T_{1}+\cdots +T_{m} \le N\}$ , where $\{T_{i}\}$ are independent copies of $T:=\text {min}\{ k : |\xi _{1}+\cdots +\xi _{k}|=2 \}$ , where $\{\xi _{i}\}$ are i.i.d. uniform $\pm 1$ variables. By Markov’s inequality, $\mathbb {P}(M<cN) \le \mathbb {P}(T_{1}+\cdots +T_{\lceil cN \rceil }>N) \le \tfrac 14$ for some universal constant $c>0$ . Thus, $\mathbb {P}(N \ge (C/c)\log n) \le \tfrac 34$ , as required.

We now turn to the second inequality in the corollary. Observe that the quantity in question is the same as $\mathbb {P}(X=0)-\mathbb {P}(X=2)$ . Conditioning on N, we have

$$ \begin{align*} \mathbb{P}(X=0 \mid N)-\mathbb{P}(X=2 \mid N) &= 2^{-N}\binom{N}{N/2} - 2^{-N}\binom{N}{N/2+1} \\&= \frac{2^{-N}}{N/2+1}\binom{N}{N/2} = \Theta(N^{-3/2}). \end{align*} $$

Using the same bounds on N as before yields the required inequality.

4.1.2 Proof of Proposition 4.1

We begin with two lemmas. The first lemma asserts that if we look at the height function inside a domain containing some box $\Lambda $ , then, with high probability, we will find a level loop of absolute height at least 2 surrounding a slightly smaller box. More precisely, the loop will be between the boundary of the domain (which could be far away $\Lambda $ ) and the boundary of a box whose size is at most a geometric number of scales smaller than $\Lambda $ .

Lemma 4.7. There exists a constant $c>0$ such that the following assertion holds. Let $k \ge 1$ , let D be a domain containing $\Lambda _{k}$ and let $(B,\kappa )$ be $|h|$ -adapted boundary conditions with $B_{\text {sym}} = \partial D$ and $B_{\text {pos}} = \emptyset $ . Let $\mathcal {E}_{i}$ be the event that the annulus $A_{2^{-i}k, 2^{-i+1}k}$ contains a level loop of $|h|\ge 2$ surrounding the origin. Then, for all $n \ge 1$ with $2^{-n}k \ge 100$ ,

$$ \begin{align*} \phi^{\kappa}_{D} (\mathcal{E}_{1} \cup \cdots \cup \mathcal{E}_{n}) \ge 1 - e^{-cn}. \end{align*} $$

Proof. Observe that $\mathcal {E}_{i}$ is measurable with respect to $|h|_{|A_{i}}$ , where $A_{i}$ is the annulus $A_{2^{-i}k, 2^{-i+1}k}$ . It suffices to show that

$$ \begin{align*} \phi^{\kappa}_{D} (\mathcal{E}_{n} \mid {\textbf{1}}_{\mathcal{E}_{1}},\ldots,{\textbf{1}}_{\mathcal{E}_{n-1}}) \ge c \quad\text{almost surely}. \end{align*} $$

In fact, we prove the stronger statement that

$$ \begin{align*} \phi^{\kappa}_{D} (\mathcal{E}_{n} \mid |h|_{|D \setminus D_{n}}) \ge c \quad\text{almost surely}, \end{align*} $$

where $D_{n}$ is the domain $\Lambda ^{\text {e}}_{2^{-n+1}k-1}$ . In words, we first explore the absolute value from the boundary of D inward until we reach the outer boundary of $A_{n}$ and their even neighbors inside $A_{n}$ , and then, regardless of what this exploration reveals, the conditional probability of $\mathcal {E}_{n}$ is at least some universal constant c.

Toward showing this, we first argue that, when sampling from $\phi ^{\kappa }_{D}$ , the conditional distribution of $|h|_{|D_{n}}$ given $|h|_{|(D \setminus D_{n}) \cup \partial D_{n}}$ is almost surely stochastically larger than $\phi ^{0}_{D_{n}}(|h| \in \cdot )$ . Indeed, the domain Markov property and the assumption that the boundary condition has $B_{\text {pos}} = \emptyset $ imply that the former conditional distribution is $\phi ^{\kappa _{n}}_{D_{n}}(|h| \in \cdot )$ , where $\kappa _{n}$ is the boundary condition on $\partial D_{n}$ that equals $\{|h_{v}|,-|h_{v}|\}$ at each $v \in \partial D_{n}$ . The FKG inequality for absolute value (Proposition 3.4) now implies that $\phi ^{\kappa _{n}}_{D_{n}}(|h| \in \cdot )$ stochastically dominates $\phi ^{0}_{D_{n}}(|h| \in \cdot )$ .

Now observe that $\mathcal {E}_{n}$ is increasing in $|h|$ so that the above stochastic domination implies that $\phi ^{\kappa }_{D}(\mathcal {E}_{n} \mid |h|_{|D \setminus D_{n}}) \ge \phi _{D_{n}}^{0}(\mathcal {E}_{n})$ almost surely. Finally, the uniform lower bound on $\phi _{D_{n}}^{0}(\mathcal {E}_{n})$ follows from (3.2) and the standard trick of using the FKG inequality (for $|h|$ ) to glue together crossings of four rectangles into a loop.

Corollary 4.8. There exists a constant $c>0$ such that for all $k \ge 1$ , all domains $D \supset \Lambda _{k}$ and all $n \ge 1$ with $2^{-n}k \ge 100$ ,

$$ \begin{align*} \phi_{D}^{0}(\Lambda_{2^{-n}k}\text{ is surrounded by a level loop of }|h|=2\text{ in }D) \ge 1-e^{-cn}. \end{align*} $$

Proof. Lemma 4.7 implies that with probability at least $1-e^{-cn}$ there is a $\times $ -loop of $|h|\ge 2$ surrounding $\Lambda _{2^{-n}k}$ in D. Since the boundary conditions put height 0 on $\partial D$ , on the former event, there must be a level loop of $|h|=2$ surrounding $\Lambda _{2^{-n}k}$ in D.

The next lemma essentially shows that a level loop of height 2 intersecting a box cannot have long ‘tentacles’ going very far away from the box. A $\times $ -crossing of an annulus inside a domain D is a $\times $ -path in D connecting the inside and the outside boundaries of the annulus.

Lemma 4.9. There exists a constant $c>0$ such that the following assertion holds. Let D be a domain and let $k \ge 100$ . For $i \ge 1$ , let $\mathcal {C}_{i}$ be the event that there exists a $\times $ -crossing of height $2$ of the annulus $A_{2^{i-1} k, 2^{i}k}$ inside D. Then, for all $n \ge 1$ ,

$$ \begin{align*}\phi_{D}^{0}(\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) \le e^{-cn}.\end{align*} $$

We remark that no assumption on the location of the domain is made. In particular, it may intersect some of the annuli without containing them.

Proof. Let $A_{i}$ be the slightly thinner even annulus $(\Lambda ^{\text {e}}_{2^{i}k-10} \setminus \Lambda ^{\text {e}}_{2^{i-1}k+10}) \cup \partial \Lambda ^{\text {e}}_{2^{i-1}k+10}$ . With a slight abuse of notation, we redefine $\mathcal {C}_{i}$ to be the event that there exists a $\times $ -crossing of height 2 of this thinner annulus inside D, noting that this increases the event so that it suffices to prove the stated inequality with this new definition.

Denote $D_{i} := A_{i} \cap D$ and $B := \bigcup _{i \ge 1} \partial D_{i}$ , and let $B_{0} := B \setminus \partial D$ be the portions of the boundaries of the even annuli in $D \setminus \partial D$ . Consider the boundary condition $(B,\kappa )$ in which $\kappa $ equals $\{0\}$ on $\partial D$ and $\{2\}$ on $B_{0}$ . We first show that

$$ \begin{align*} \phi^{0}_{D}(\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) \le \phi_{D}^{B,\kappa} (\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}).\end{align*} $$

Let $\mathcal {C}^{\prime }_{i}$ be the event that there exists a $\times $ -crossing of height $0$ of the annulus $A_{i}$ inside D. Note that, by symmetry,

$$ \begin{align*} \phi^{0}_{D}(\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) = \phi^{-2}_{D}(\mathcal{C}^{\prime}_{1} \cap \cdots \cap \mathcal{C}^{\prime}_{n}) = \phi^{2}_{D}(\mathcal{C}^{\prime}_{1} \cap \cdots \cap \mathcal{C}^{\prime}_{n}) \end{align*} $$

and, similarly,

$$ \begin{align*} \phi^{B,\kappa}_{D}(\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) = \phi^{B,\kappa^{\prime}}_{D}(\mathcal{C}^{\prime}_{1} \cap \cdots \cap \mathcal{C}^{\prime}_{n}),\end{align*} $$

where $\kappa ^{\prime }$ equals $\{2\}$ on $\partial D$ and $\{0\}$ on $B_{0}$ . Thus, it suffices to show that

$$ \begin{align*} \phi^{2}_{D}(\mathcal{C}^{\prime}_{1} \cap \cdots \cap \mathcal{C}^{\prime}_{n}) \le \phi_{D}^{B,\kappa^{\prime}} (\mathcal{C}^{\prime}_{1} \cap \cdots \cap \mathcal{C}^{\prime}_{n}).\end{align*} $$

Since $\mathcal {C}^{\prime }_{i}$ is a decreasing event in the absolute value of h, this follows from the FKG inequality for absolute value (Proposition 3.4), where the constant boundary condition 2 is viewed as the boundary condition $(B,\kappa ^{\prime \prime })$ in which $\kappa ^{\prime \prime }$ equals $\{2\}$ on $\partial D$ and $\mathbb {Z}$ on $B_{0}$ (the latter is equivalent to having no condition on the height on $B_{0}$ , that is, vertices on $B_{0}$ are free to take any value). Here, both boundary conditions have the same partition of B into $B_{\text {pos}}=\partial D$ and $B_{\text {sym}}=B_{0}$ .

Observe that the boundary condition $(B,\kappa )$ decouples the events $\mathcal {C}_{1},\ldots ,\mathcal {C}_{n}$ . Thus,

$$ \begin{align*} \phi^{0}_{D}(\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) \le \phi_{D}^{B,\kappa} (\mathcal{C}_{1} \cap \cdots \cap \mathcal{C}_{n}) &= \phi_{D}^{B,\kappa}(\mathcal{C}_{1}) \cdots \phi_{D}^{B,\kappa}(\mathcal{C}_{n})\\ &= \phi_{D_{1}}^{\kappa_{1}}(\mathcal{C}_{1}) \cdots \phi_{D_{n}}^{\kappa_{n}}(\mathcal{C}_{n}),\end{align*} $$

where $\kappa _{i}$ equals $\{0\}$ on $\partial D_{i} \cap \partial D$ and $\{2\}$ on $\partial D_{i} \setminus \partial D$ . It remains to show that

$$ \begin{align*} \phi_{D_{i}}^{\kappa_{i}}(\mathcal{C}_{i}) \le 1-\alpha \end{align*} $$

for some universal constant $\alpha>0$ . The case when $D_{i}=A_{i}$ (that is, when the domain D contains the annulus $A_{i}$ ) is more straightforward than the case $D_{i} \subsetneq A_{i}$ . In order to treat the latter case in a similar way to the former (and simultaneously), it is convenient to work in the entire annulus $A_{i}$ rather than in the portion $D_{i}$ of it. To this end, with a slight abuse of notation, we extend $\kappa _{i}$ to $B_{i}=\partial D_{i} \cup \partial A_{i}$ by defining it to be $\{2\}$ on $\partial A_{i} \setminus \partial D_{i}$ . Then, by the domain Markov property, $\phi _{D_{i}}^{\kappa _{i}}(\mathcal {C}_{i}) = \phi _{A_{i}}^{B_{i},\kappa _{i}}(\mathcal {C}_{i})$ . Now consider the event $\bar {\mathcal {C}}_{i}$ that there exists a $\times $ -crossing of height $2$ of the annulus $A_{i}$ (not necessarily inside D). Clearly, $\mathcal {C}_{i} \subset \bar {\mathcal {C}}_{i}$ so that $\phi _{A_{i}}^{B_{i},\kappa _{i}}(\mathcal {C}_{i}) \le \phi _{A_{i}}^{B_{i},\kappa _{i}}(\bar {\mathcal {C}}_{i})$ . Let $\mathcal {E}_{i}$ be the event that there exists a $*$ -loop of height equal to 0 surrounding the origin inside $A_{i}$ (a $*$ -path is a sequence of vertices in $\mathbb {Z}^{2}$ with consecutive vertices at nearest-neighbor distance equal to 2). By duality, the events $\bar {\mathcal {C}}_{i}$ and $\mathcal {E}_{i}$ are disjoint. Indeed, by [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9, Lemma 2.4, first item], a $\times $ -cluster of height at least $2$ is blocked by a $*$ -loop of height at most 0. Since, however, the boundary conditions on $\partial A_{i}$ are at least 0 everywhere (namely, 0 or 2), the latter is equivalent to having a $*$ -loop of height exactly equal to 0 in the annulus. Thus, $\phi _{A_{i}}^{B_{i},\kappa _{i}}(\bar {\mathcal {C}}_{i}) \le 1 - \phi _{A_{i}}^{B_{i},\kappa _{i}}(\mathcal {E}_{i})$ . So far, we have shown that

$$ \begin{align*} \phi_{D_{i}}^{\kappa_{i}}(\mathcal{C}_{i}) \le 1 - \phi_{A_{i}}^{B_{i},\kappa_{i}}(\mathcal{E}_{i}).\end{align*} $$

It remains to show that $\phi _{A_{i}}^{B_{i},\kappa _{i}}(\mathcal {E}_{i})$ is uniformly bounded below. To this end, we first argue that $\phi _{A_{i}}^{B_{i},\kappa _{i}}(\mathcal {E}_{i}) \ge \phi _{A_{i}}^{2}(\mathcal {E}_{i})$ . Indeed, since the event $\mathcal {E}_{i}$ is decreasing in $|h|$ , by viewing the constant two-boundary condition as equaling $\mathbb {Z}$ on $B_{i} \setminus \partial A_{i}$ (that is, taking free boundary conditions there), this follows from the FKG inequality for absolute value. Finally, the estimate $\phi _{A_{i}}^{2}(\mathcal {E}_{i}) \ge \alpha $ follows from (3.2) and the standard trick of using the FKG inequality (again, for $|h|$ ) to glue together crossings of four rectangles into a loop.

We are now ready to prove Proposition 4.1.

Proof of Proposition 4.1

For $j>i \ge 0$ , let $N_{i,j}$ be the number of essential level loops contained in $A_{2^{i}k,2^{j}k}$ and let $N^{\prime }_{i,j} \ge N_{i,j}$ be the number of essential level loops intersecting $A_{2^{i}k,2^{j}k}$ . We are interested in lower-bounding $N=N_{0,a}$ and upper-bounding $N^{\prime }=N^{\prime }_{0,a}$ . For the reader’s convenience, we recall the precise statements in the proof below.

Upper bound. We show the upper bound (4.1) on $N^{\prime }$ , namely, that for some universal constants $C,c>0$ , we have

(4.3) $$ \begin{align} \phi_{D}^{0}(N^{\prime} \ge n) \le e^{-cn \log(\frac na \wedge k)} \quad\text{for any }n \ge Ca. \end{align} $$

The proof follows a strategy similar to that of [Reference Duminil-Copin, Harel, Laslier, Raoufi and Ray9, Proposition 4.8] with some simplifications. Let $D_{m}$ be the domain enclosed by $\mathcal {L}_{m}$ and recall that $H_{m}$ is the height on $\mathcal {L}_{m}$ . Consider the line $(k,2^{a}k] \times \{0\}$ crossing the annulus $A_{k,2^{a}k}$ . Let $\ell _{m} \in (k,2^{a}k]$ denote the distance to the origin of the rightmost intersection point $(\ell _{m},0)$ of $\mathcal {L}_{m}$ with this line. Fix a large r, and let $\mathcal {B}_{m}$ be the event that $\ell _{m}-\ell _{m+1} \le 2^{-r}\ell _{m}$ , or equivalently, $\ell _{m+1} \ge (1-2^{-r})\ell _{m}$ . In particular, on this event, the loops $\mathcal {L}_{m}$ and $\mathcal {L}_{m+1}$ are relatively close to one another. Specifically, if $\mathcal {B}_{m}$ occurs, then the loop $\mathcal {L}_{m+1}$ creates a crossing of $A_{2^{-r}\ell _{m},\ell _{m}} ((\ell _{m},0)) \cap D_{m}$ by a $\times $ -path of height $H_{m+1}$ . Thus, by Lemma 4.9,

$$ \begin{align*} \phi_{D}^{0}(\mathcal{B}_{m} \mid \mathcal{L}_{1},\ldots,\mathcal{L}_{m}) \le e^{-dr},\end{align*} $$

where $d>0$ is a universal constant, as long as $2^{-r}\ell _{m} \ge 100$ .

Since $\mathcal {B}_{m}$ occurs unless $\log \ell _{m} - \log \ell _{m+1}> -\log (1-2^{-r}) \ge 2^{-r}$ , it is straightforward that all but at most $2^{r}a \log 2$ of the essential level loops $\mathcal {L}_{m}$ intersecting $A_{k,2^{a}k}$ must trigger the corresponding event $\mathcal {B}_{m}$ . Thus, by the above bound on the conditional probability of $\mathcal {B}_{m}$ and a union bound on those m for which $\mathcal {B}_{m}$ occurs, we have

$$ \begin{align*} \phi_{D}^{0} (N^{\prime} = n ) \le 2^{n} e^{-dr(n-2^{r}a \log 2)}. \end{align*} $$

Now choosing r to be the minimum between $\log _{2} ( n/{2a})$ and $\log _{2}({k}/{100})$ (so as to ensure that $2^{-r}\ell _{m} \ge 100$ always holds) and summing over n yields the required bound (4.3).

Lower bound. We now turn to the lower bound (4.2) on $N $ , namely, that for some universal constants $C,c>0$ , if D contains $\Lambda _{2^{a}k}$ , then

$$ \begin{align*} \phi_{D}^{0}(N \le ca) \le Ce^{-ca}.\end{align*} $$

The main issue to overcome is the case when D is much further away from the annulus in question. We need to get around the fact that the outermost essential level loop which intersects $\Lambda _{2^{a}k}$ is not too irregular. We divide the proof into three steps.

Step 1. We first establish a similar bound on the number of essential level loops surrounding $\Lambda _{k}$ . Namely,

(4.4) $$ \begin{align} \phi_{D}^{0}(N_{0,\infty} \le ca) \le e^{-ca}. \end{align} $$

For $0 \le m < N_{0,\infty }$ , let $S_{m}$ denote the largest integer $i \ge 0$ such that $\mathcal {L}_{m}$ surrounds $\Lambda _{2^{i}k}$ . We think of $S_{m}$ as the scale at which we fully discover $\mathcal {L}_{m}$ . Set $S_{m}=-1$ for $m \ge N_{0,\infty }$ . Note that $(S_{m})_{m=0}^{\infty }$ is a decreasing sequence with $S_{0} \ge a$ and $S_{N_{0,\infty }-1} \ge 0$ . By Corollary 4.8, for any $m \ge 0$ , the conditional distribution of $S_{m}-S_{m+1}$ given $\mathcal {L}_{1},\ldots ,\mathcal {L}_{m}$ is almost surely stochastically dominated by a geometric random variable of some universal parameter p. Thus,

$$ \begin{align*} N_{0,\infty}\text{ stochastically dominates }1+\max \{ m \ge 0 : \xi_{1}+\cdots+\xi_{m} \le a\},\end{align*} $$

where $\{\xi _{i}\}$ are independent geometric random variables with parameter p. It now follows from a standard Chernoff bound that there exists a universal constant $c>0$ such that

$$ \begin{align*} \phi_{D}^{0}(N_{0,\infty} \le ca) \le \mathbb{P}(\xi_{1}+\cdots+\xi_{\lfloor ca \rfloor}> a) \le e^{-ca}.\end{align*} $$

Step 2. Next, we establish the lower bound on N in the case when the domain D does not contain $\Lambda _{2^{a+1}k}$ . In this case, any essential level loop not contained in $\Lambda _{2^{a}k}$ must intersect $A_{2^{a}k,2^{a+1}k}$ , so that $N=N_{0,\infty } - N^{\prime }_{a,a+1}$ . Thus,

$$ \begin{align*} \phi_{D}^{0}\bigg(N \le \frac{ca}2\bigg) \le \phi_{D}^{0}(N_{0,\infty} \le ca) + \phi_{D}^{0}\bigg(N^{\prime}_{a,a+1} \ge \frac{ca}2\bigg) \le Ce^{-ca}, \end{align*} $$

where the last inequality follows from (4.4) and (4.3). We note for use in the next step that this inequality can be restated (by applying it with a different choice of k and a, and readjusting the constants appropriately) as, for any $a_{2} \ge a_{1} \ge 0$ and any domain D such that $\Lambda _{2^{a_{2}}k} \subset D \not \supset \Lambda _{2^{a_{2}+1}k}$ ,

(4.5) $$ \begin{align} \phi_{D}^{0}(N_{a_{1},a_{2}} \le c(a_{2}-a_{1})) \le Ce^{-c(a_{2}-a_{1})}. \end{align} $$

Step 3. Finally, we are ready to establish the bound for an arbitrary domain D containing $\Lambda _{2^{a}k}$ . Let I be the smallest integer $i>a/2$ such that there exists an essential level loop surrounding $\Lambda _{2^{a/2}k}$ and intersecting $\Lambda _{2^{i+1}k}$ (recall that we also consider the boundary of the domain as an essential level loop, so that I is always finite). It suffices to show that

$$ \begin{align*} \phi_{D}^{0}(N \le ca,~I=i) \le Ce^{-ci} \quad\text{for all }i>a/2.\end{align*} $$

Fix $i>a/2$ and suppose that the event $\{I=i\}$ occurs. Let $M_{0}$ denote the smallest index $0 \le m \le M$ such that the loop $\mathcal {L}_{m}$ intersects $\Lambda _{2^{i+1}k}$ , and note that this loop necessarily surrounds $\Lambda _{2^{i}k}$ . Let $D_{0}$ be the domain enclosed by $\mathcal {L}_{M_{0}}$ , and note that $D_{0}$ contains $\Lambda _{2^{i}k}$ but not $\Lambda _{2^{i+1}k}$ . Note also that we may explore $\mathcal {L}_{1},\ldots ,\mathcal {L}_{M_{0}}$ from the outside, so that the height function inside $D_{0}$ is conditionally independent by the domain Markov property. We now consider two cases: if $i \le a$ , then $N \ge N_{0,i}$ , so that (4.5) yields the required bound; and if $i>a$ , then $N_{a/2,i}=0$ , so that (4.5) again yields the required bound.

4.2 Proof of Theorem 3.1

Our goal is to show that the family $\mathcal {M}$ of all pairs $(\mu ^{ij}_{D},D)$ , with D finite and simply connected and $i,j \in \{0,1,2\}$ distinct, is locally mixing with a power-law rate function. To this end, we fix $n \ge 1$ , distinct $i,j \in \{0,1,2\}$ , distinct $i^{\prime },j^{\prime } \in \{0,1,2\}$ and two finite and simply connected sets $D,D^{\prime } \subset \mathbb {Z}^{2}$ containing $\Lambda _{n}$ , and we aim to construct a coupling between $f \sim \mu ^{ij}_{D}$ and $f^{\prime } \sim \mu ^{i^{\prime }j^{\prime }}_{D^{\prime }}$ with the desired properties (recall Definition 2.1). We first handle the case when D and $D^{\prime }$ are domains and $i=i^{\prime }=0$ (in  which case j and $j^{\prime }$ are irrelevant), and later explain the general case. We further assume without loss of generality that one of the domains, say $D^{\prime }$ , is $\Lambda _{n}^{\text {e}}$ , in which case the other domain necessarily contains it (the required coupling is then easily obtained from two such couplings).

In order not to introduce cumbersome notation, we begin by describing the coupling ‘as seen from’ a single vertex v. The construction will involve an iteration of a three-step procedure in which we alternate between exploring (1) only f, (2) both f and $f^{\prime }$ simultaneously, and (3) only $f^{\prime }$ . At the end of iteration i, we will have defined a sequence $L_{0},L^{\prime }_{0},L_{1},L^{\prime }_{1},\ldots ,L_{i},L^{\prime }_{i}$ of nested even $\times $ -loops surrounding v (some loops may coincide or partially overlap), where $L_{0},\ldots ,L_{i}$ are color-0 loops of f and $L^{\prime }_{0},\ldots ,L^{\prime }_{i}$ are color-0 loops of $f^{\prime }$ , and we will have fully explored f and $f^{\prime }$ outside of $L_{i}$ and $L^{\prime }_{i}$ , respectively, but not at all inside the domains $D_{i}$ and $D^{\prime }_{i}$ enclosed by them; see Figure 1. In particular, the domain Markov property will imply that at the end of iteration i, the conditional distributions of $f_{|D_{i}}$ and $f^{\prime }_{|D^{\prime }_{i}}$ will be $\mu ^{0}_{D_{i}}$ and $\mu ^{0}_{D^{\prime }_{i}}$ , respectively.

Figure 1 The sequence of nested loops $L_{0},L^{\prime }_{0},L_{1},L^{\prime }_{1},\ldots $ surrounding v.

Initially, we set $L_{0}:=\partial D$ and $L^{\prime }_{0}:=\partial D^{\prime }$ , and note that $L_{0}$ surrounds $L^{\prime }_{0}$ by our assumption on the domains. This is all that is done in iteration 0. Suppose that we have completed iteration i and that the conditional distributions of $f_{|D_{i}}$ and $f^{\prime }_{|D^{\prime }_{i}}$ are $\mu ^{0}_{D_{i}}$ and $\mu ^{0}_{D^{\prime }_{i}}$ . Let us now explain how to define $L_{i+1}$ and $L^{\prime }_{i+1}$ . At this point, we switch to the height function representation, noting that we may view $f_{|D_{i}}$ as a height function $h_{i}$ for which $L_{i}$ is a level loop of height 0, and similarly, we may view $f^{\prime }_{|D^{\prime }_{i}}$ as a height function $h^{\prime }_{i}$ for which $L^{\prime }_{i}$ is a level loop of height 0. Thus, $h_{i} \sim \phi ^{0}_{D_{i}}$ and $h^{\prime }_{i}\sim \phi ^{0}_{D^{\prime }_{i}}$ . We now use this representation in order to describe the three-step exploration procedure of f and $f^{\prime }$ in iteration $i+1$ . For readability, we write $h=h_{i}$ and $h^{\prime }=h^{\prime }_{i}$ below.

Step 1. We explore the absolute value $|h|$ of the height function h (independently of $h^{\prime }$ ) up to the loop $L^{\prime }_{i}$ (that is, on $D_{i} \setminus D^{\prime }_{i}$ and on the boundary of $D^{\prime }_{i}$ ), thereby revealing a random $|h|$ -adapted boundary condition for h on the domain $D^{\prime }_{i}$ . By the domain Markov property, the conditional law of h in $D^{\prime }_{i}$ is that of a uniform homomorphism height function in $D^{\prime }_{i}$ with this boundary condition (which has $B_{\text {sym}}=L^{\prime }_{i}$ and $B_{\text {pos}}=\emptyset $ ).

Step 2. Since $h^{\prime }$ is zero on $L^{\prime }_{i}$ , the FKG inequality for absolute value implies that $|h|_{|D^{\prime }_{i}}$ stochastically dominates $|h^{\prime }|_{|D^{\prime }_{i}}$ (given the exploration in the first step). Consequently, we can construct a coupling between $|h|$ and $|h^{\prime }|$ inside $D^{\prime }_{i}$ by simultaneously exploring and revealing the values of both $|h|$ and $|h^{\prime }|$ , vertex by vertex, starting from the boundary of $D^{\prime }_{i}$ inwards, ensuring along the way that $|h| \ge |h^{\prime }|$ (this type of exploration is standard; see, for example, [Reference van den Berg and Maes35]). We explore $|h|$ and $|h^{\prime }|$ in this way until we discover the outermost level loop of height 0 mod 6 for h surrounding v and inside $D^{\prime }_{i}$ ; this loop is $L_{i+1}$ (note that it is a color-0 even $\times $ -loop for f).

Step 3. The domain Markov property implies that, at this point, the conditional distributions of $|h|_{|D_{i+1}}$ and $|h^{\prime }|_{|D_{i+1}}$ are those of the absolute values of uniform homomorphism height functions in $D_{i+1}$ with the corresponding $|h|$ -adapted boundary conditions that have been revealed by the exploration in the previous step. If these boundary conditions happen to be identical, then we jointly sample h and $h^{\prime }$ in the domain $D_{i+1}$ according to a single sample from their common distribution, and we stop the iterative procedure; in this case, we say that the $i+1$ iteration resulted in a successful coupling for v. Otherwise, we continue exploring $|h^{\prime }|$ alone (independently of h) until we discover the outermost level loop of height 0 mod 6 for $h^{\prime }$ surrounding v and inside $D_{i+1}$ ; this loop is $L^{\prime }_{i+1}$ . At this point, we have coupled the absolute values of the height functions up to $L_{i+1}$ and $L^{\prime }_{i+1}$ . Finally, we complete the coupling h and $h^{\prime }$ by coupling their signs in any manner (for example, independently).

This completes the description of iteration $i+1$ , at the end of which we have explored f up to $L_{i+1}$ and $f^{\prime }$ up to $L^{\prime }_{i+1}$ , so that by the domain Markov property, the conditional distributions of $f_{|D_{i+1}}$ and $f^{\prime }_{|D^{\prime }_{i+1}}$ are indeed $\mu ^{0}_{D_{i+1}}$ and $\mu ^{0}_{D^{\prime }_{i+1}}$ , as claimed. If at some iteration i we cannot find one of the loops we are looking for (that is, $L_{i}$ or $L^{\prime }_{i}$ does not exist), then we stop the iterative procedure and say that the coupling has failed for v. We emphasize that in iteration i, when we discover the loops $L_{i}$ and $L^{\prime }_{i}$ , the heights of h and $h^{\prime }$ on these loops are some (different) multiples of 6, but that in the next iteration we then shift each of the two height functions by the corresponding amount so that the height again becomes 0 for both before we start looking for the loops $L_{i+1}$ and $L^{\prime }_{i+1}$ .

The actual coupling between f and $f^{\prime }$ does not treat v as a distinguished vertex, but rather attempts to couple all vertices in $\Lambda _{n}$ in parallel. Specifically, whenever we tried to find the outermost level loop of height 0 mod 6 surrounding v, we instead find all outermost level loops of height 0 mod 6 (with no specific target vertex). The collection of all such loops is still explorable from the outside. Thus, iteration i generates for us two collections of loops $\{L_{i,j}\}_{j}$ and $\{L^{\prime }_{i,j,j^{\prime }}\}_{j,j^{\prime }}$ (with $L^{\prime }_{i,j,j^{\prime }}$ nested inside $L_{i,j}$ ), and in iteration $i+1$ we recursively repeat this procedure inside each of these loops. This completes the description of the coupling between f and $f^{\prime }$ .

We now turn to show that the constructed coupling has the required properties. We first check that $f_{|D \setminus \Lambda _{n}}$ and $f^{\prime }$ are independent. Indeed, in the first step of the first iteration of the construction, we explore f independently of $f^{\prime }$ up to $\partial D^{\prime }$ and thereby reveal $f_{|(D \setminus D^{\prime }) \cup \partial D^{\prime }}$ . Since $D^{\prime } \setminus \partial D^{\prime } \subset \Lambda _{n}$ and since, conditionally on this exploration, the law of $f^{\prime }$ is still $\mu ^{0}_{D^{\prime }}$ , we see that $f^{\prime }$ is independent of $f_{|D \setminus \Lambda _{n}}$ .

We now turn to the main issue at hand, namely, to show that the constructed coupling has a good chance of successfully coupling any given vertex. To be precise, we need to show that, under this coupling, $\mathbb {P}(f(v) \neq f^{\prime }(v)) \le C(n-k)^{-\alpha }$ for some $C,\alpha>0$ and any $0 \le k \le n$ and $v \in \Lambda _{k}$ . Fix such a k and v. By construction, $f(v)$ and $f^{\prime }(v)$ are equal unless the coupling fails for v. Thus, it suffices to show that

(4.6) $$ \begin{align} \mathbb{P}(\text{coupling fails for }v) \le C(n-k)^{-\alpha}. \end{align} $$

We continue to use the loops $L_{i}$ and $L^{\prime }_{i}$ as defined above with respect to v. Let $\mathcal {F}_{i}$ denote the $\sigma $ -algebra generated by $L_{0},L^{\prime }_{0},\ldots ,L_{i},L^{\prime }_{i}$ and $f_{|(D \setminus D_{i}) \cup L_{i}}$ and $f^{\prime }_{|(D^{\prime } \setminus D^{\prime }_{i}) \cup L^{\prime }_{i}}$ . Note that $\mathcal {F}_{i}$ represents the information revealed at the end of iteration i.

Let us first show that it is unlikely that the coupling fails for v after few iterations. To be precise, we claim that for some constants $C,c,\alpha>0$ , we have

(4.7) $$ \begin{align} \mathbb{P}(\text{coupling fails for }{v}\text{ before iteration }c \log(n-k)) \le C(n-k)^{-\alpha}. \end{align} $$

To this end, let $S_{i}$ denote the largest $j \ge 0$ such that $L^{\prime }_{i}$ surrounds $\Lambda _{2^{j}}(v)$ , and set $S_{i}=-1$ if $L^{\prime }_{i}$ does not exist. Note that $S_{0} \ge S_{1} \ge S_{2} \ge \cdots $ since the loops are nested, and that $S_{0} \ge \lfloor \log _{2}(n-k) \rfloor $ since $\Lambda _{n-k}(v) \subset \Lambda _{n} \subset D^{\prime }$ . Note also that if the coupling fails for v before iteration i, then $S_{i}=-1$ . Thus, it suffices to bound $\mathbb {P}(S_{m}=-1)$ , where $m := \lceil c\log (n-k) \rceil $ . We claim that, for any $i \ge 0$ , conditioned on $\mathcal {F}_{i}$ , the difference $S_{i}-S_{i+1}$ is almost surely stochastically dominated by a random variable T having exponential tails. Indeed, if $S_{i}=s$ and $S_{i}-S_{i+1} \ge 2t+1$ , then $t \le s/2$ and either the annulus $A_{2^{s-t},2^{s}}(v)$ contains no loop of height 0 mod 6 surrounding v for $h_{i}$ or the annulus $A_{2^{s-2t},2^{s-t}}(v)$ contains no loop of height 0 mod 6 surrounding v for $h^{\prime }_{i}$ . Corollary 4.2 implies that, given $\mathcal {F}_{i}$ , each of these events has probability at most $Ce^{-ct}$ for some universal constants $C,c>0$ . Therefore, letting $\{T_{j}\}_{j}$ be independent copies of T,

$$ \begin{align*} \mathbb{P}(S_{m}=-1) \le \mathbb{P}(T_{1}+\cdots+T_{m} \ge \lfloor \log_{2}(n-k) \rfloor) \le C(n-k)^{-\alpha},\end{align*} $$

where the second inequality follows from a standard Chernoff bound for i.i.d. random variables with exponential tails by choosing c small enough.

Now that we know that the coupling does not fail for v before order $\log (n-k)$ iterations, we aim to show that in each such iteration there is a constant probability of a successful coupling for v. It will be helpful to consider two consecutive iterations at a time. Thus, we aim to show that, for some constants $K,c>0$ , for any even $i \ge 0$ , conditioned on $\mathcal {F}_{i}$ , on the event that $L^{\prime }_{i}$ is at distance at least K from v, with probability at least c, the $i+2$ iteration results in a successful coupling for v. This yields that, for all $i \ge 0$ ,

$$ \begin{align*} \mathbb{P}(\text{coupling neither succeeds nor fails for }{v}\text{ before iteration }i) \le C(1-c)^{i}.\end{align*} $$

Together with (4.7), this gives the required bound (4.6). The reason for considering two consecutive iterations is that, given $\mathcal {F}_{i}$ , it may happen that $L^{\prime }_{i}$ is deep inside $L_{i}$ . We use the first of the two iterations to gain some control on this, showing that, regardless of the relative geometry of $L_{i}$ and $L^{\prime }_{i}$ , there is a constant probability that $L^{\prime }_{i+1}$ is not far from $L_{i+1}$ . In the next iteration, conditioning on $\mathcal {F}_{i+1}$ , we may then assume that we are on this good event, in which case we will be able to show that there is a constant probability that $L_{i+2}$ is a color-0 loop for both f and $f^{\prime }$ (in fact, a loop of height 0 for both $h_{i+1}$ and $h^{\prime }_{i+1}$ ), resulting in a successful coupling for v. We now make this precise.

Condition on $\mathcal {F}_{i}$ and consider the loops $L_{i}$ and $L^{\prime }_{i}$ . By what we have shown above about $S_{i}-S_{i+1}$ , with probability at least $\frac {1}{2}$ , we have that

(4.8) $$ \begin{align} \Lambda_{2^{s-a}}(v) \subset D^{\prime}_{i+1} \subset D_{i+1} \subset D^{\prime}_{i} \not\supset \Lambda_{2^{s+1}}(v) \end{align} $$

for some universal constant a, where $s:=S_{i}$ . Now condition on $\mathcal {F}_{i+1}$ and assume that (4.8) occurs. Then Corollary 4.3 implies that $L_{i+2}$ exists and is a level loop of height 0 for $h_{i+1}$ with probability at least c for some universal constant $c>0$ , as long as $2^{s-a}$ is larger than some universal constant $K^{\prime }$ (which is ensured by choosing $K=2^{a+1}K^{\prime }$ ). Now observe that the domination maintained in the second step of the construction of the coupling implies that $L_{i+2}$ must also be a level loop of height 0 for $h^{\prime }_{i+1}$ , implying that iteration $i+2$ resulted in a successful coupling for v. Thus, there is probability at least $c/2$ that iteration $i+2$ results in a successful coupling for v. This finishes the proof that the constructed coupling has the two properties required by the local mixing condition.

We have shown above that the family $\mathcal {M}^{\prime } \subset \mathcal {M}$ consisting of all pairs $(\mu ^{0}_{D},D)$ , where D is a domain, is locally mixing with a power-law rate function. It remains to explain that $\mathcal {M}$ is locally mixing with such a rate function. Suppose that $f \sim \mu ^{ij}_{D}$ for general $D,i,j$ . We may still assume as before that $D^{\prime }$ is the domain $\Lambda ^{\text {e}}_{n}$ and that $i^{\prime }=0$ . The above proof applies to this situation as is, with the only difference being that $L_{0}$ is no longer a color-0 loop for f so that when we appeal to Corollary 4.2 in the first iteration (that is, when arguing that $L_{1}$ is discovered quickly), we need to use the full strength of the corollary (the moreover part). In fact, this shows more, namely, that the larger class $\mathcal {M}^{\prime \prime } \supset \mathcal {M}$ consisting of all pairs $(\mu ^{\xi }_{D},D)$ with D finite and simply connected and $\xi $ a feasible boundary condition with bounded oscillation (as in the sense of Remark 4.5) is also locally mixing with a power-law rate function.