The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

We study Swendsen--Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\textrm{mix}}$ is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large $q$, where the phase transition is discontinuous, the authors recently showed that $t_{\textrm{mix}}$ is highly sensitive to boundary conditions: $t_{\textrm{mix}} \geq \exp(cn)$ on an $n\times n$ box with periodic boundary, yet under free or monochromatic boundary conditions, $t_{\textrm{mix}} \leq\exp(n^{o(1)})$. In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the $q$ colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce $t_{\textrm{mix}} \geq \exp(cn)$, yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.


Introduction
The q-state Potts model at inverse temperature β > 0 is a generalization of the Ising model (q = 2) to q ≥ 3 possible states. It is a canonical model of statistical physics and is one of the simplest models exhibiting a discontinuous (first-order) phase transition for some choices of q. Concretely, the model on a graph G is a probability distribution over {1, ..., q} V (G) with µ(σ) ∝ exp(β ij∈E(G) 1{σ i = σ j }). Much of the analysis of the Potts model relies heavily on the random cluster (FK) model; the FK model is a model of dependent bond percolation parametrized by (p, q), identified with the q-state Potts model via the Edwards-Sokal coupling [8] when q is integer and p = 1 − e −β .
On Z 2 , substantial recent progress has been made in understanding the Potts and random cluster phase transitions in β and p, respectively. On that geometry, the critical p c (q) = 1 − e −βc was identified [1] for all q ≥ 1 with the self-dual point p sd (q) = √ q 1+ √ q . It was shown in [7] that for q ≤ 4, the phase transition is continuous (there is a unique infinite-volume Gibbs measure at β c (q)) whereas for q > 4, the phase transition is discontinuous [6] (there are q+1 extremal infinite-volume Gibbs measures corresponding to q ordered phases and an additional disordered phase). In the latter case, the phase asymmetry at the critical point is expected to induce order-order and order-disorder surface tensions (known rigorously for q large [14,19]). In the present work we study the relationship between the order-disorder surface tension and the effect of boundary conditions on mixing times (time to reach equilibrium) for the critical 2D Potts model. Specifically, we study the Swendsen-Wang dynamics [22], a non-local Markov chain suggested in the physics literature as a fast MCMC sampler of the Potts model, as it switches between the different ordered phases (low-temperature bottlenecks) by moving through the FK representation of the Potts model using the Edwards-Sokal coupling. The authors of [10] analyzed the mixing times of the Swendsen-Wang dynamics at β c (q) on subsets of Z 2 as the parameter q varied. In [10], polynomial and quasipolynomial upper bounds independent of the boundary conditions were proved for q ≤ 4. When q > 4, it was shown that t mix ≥ exp(cn) on the n × n torus (where the dynamics is slowed by the free energy barrier between the disordered phase and the q ordered phases), while on the n × n box with free or red boundary conditions and large q, the authors proved that t mix ≤ exp(n o (1) ).
This sensitivity to boundary conditions is in analogy to the sensitivity of mixing to boundary conditions in the low-temperature Ising Glauber dynamics, where for low enough temperatures, the plus-minus surface tension is very well understood [5] via cluster expansion. There, in Z 2 , the first sub-exponential bound of t mix ≤ exp(n 1 2 +o(1) ) was obtained in [17] for all sufficiently low temperatures under plus boundary. This was improved to t mix ≤ exp(n o(1) ) in [18] and subsequently to t mix ≤ n O(log n) for all β > β c in [16]; it is believed to be of order n 2 , governed by motion by mean-curvature (cf. [12]). However, at β c , all known bounds are independent of the boundary conditions; in fact, it is believed that t mix ≍ n z for some universal constant z. Such behavior should hold through q ≤ 4. (See, e.g., [9,10] for a more extensive account of related literature.) When q > 4 is sufficiently large, similar cluster expansion techniques (large β is now replaced by large q) lead to an emergent order-disorder surface tension at the critical point, destroying the independence of mixing times and boundary conditions. Certain boundary conditions can destabilize the order-disorder phase symmetry, eliminating the exponential bottlenecks in the state space. Thus, when the boundary conditions are monochromatic or free, the mixing time of Swendsen-Wang was shown [10] to be exp(n o (1) ) and is believed to be n O(1) , matching the above picture for the low temperature Ising model under plus boundary conditions.
In the present paper, we investigate the relationship between mixing times and boundary conditions that interpolate between periodic and free/monochromatic, at the critical point of a discontinuous phase transition (see Figure 1). Our results hold for q large enough, and are expected to hold whenever the order-disorder surface tension (see Def. 2.1) is positive, which in turn should hold for all q > 4.
For two sequences f n , g n , here and throughout the paper, we say f n g n if there exists C > 0 such that f n ≤ Cg n for all n and analogously define f n g n . We let Λ n,m = ([0, n]×[0, m])∩Z 2 with nearest-neighbor edges. The boundary of Λ n,m , denoted ∂Λ n,m , is the set of vertices of Λ n,m adjacent to Z 2 \ Λ n,m ; let ∂ n Λ = ∂Λ n,m ∩ (Z × {m}) be the its northern boundary of Λ, and define ∂ e Λ, ∂ w Λ, ∂ s Λ similarly.
Theorem 1 (Mixed b.c.). Let q be large, ε > 0, and let (a n , b n , c n , d n ) be marked vertices on ∂Λ n,n such that they are not all within εn of any one side of ∂Λ n,n and are all distance greater than εn from each other. There exists c(ε, q) > 0 (independent of n and a n , b n , c n , d n ) such that the Swendsen-Wang dynamics on Λ n,n at β = β c (q) with boundary conditions that are red on the boundary segments (a n , b n ) and (c n , d n ) and free elsewhere, has t mix exp(cn) . In particular, this holds with red boundary conditions on ∂ e,w Λ n,n and free elsewhere.
Theorem 2 (Dobrushin b.c.). Let q be large and let (a n , b n ) be marked vertices on ∂Λ n,n . There exists c(q) > 0 (independent of n and a n , b n ) so that Swendsen-Wang dynamics at β = β c (q) with boundary conditions that are red on the boundary segment (a n , b n ) and free elsewhere, has t mix exp(c n log n) . In particular, this holds with red boundary conditions on ∂ s,w Λ n,n and free elsewhere.
Theorem 3 (Cylinders). Let q be large. The critical Swendsen-Wang dynamics with periodic boundary conditions on ∂ n,s Λ n,n and either red or free boundary conditions on each of ∂ e Λ n,n and ∂ w Λ n,n satisfies 4

REZA GHEISSARI AND EYAL LUBETZKY
The Potts and FK models. Define the q-state Potts model on a graph G = (V, E) as the probability measure µ G,β,q on Ω P = [q] V := {1, ..., q} V as P is the partition function. Define the random cluster (FK) model on a graph G as the probability measure π G,p,q on state space Ω fk = {0, 1} E as where o(ω) = e∈E ω(e) and k(ω) is the number of connected components (clusters) in the subgraph of G induced by ω (we count singletons as their own clusters). We call edges that have ω(e) = 1 open, or wired, and edges that have ω(e) = 0, closed, or free.
Potts and FK boundary conditions. Consider the Potts and FK models on G = (V, E) with boundary ∂G ⊂ V . A Potts boundary condition on ∂G ⊂ G is an assignment of spin values η ∈ [q] ∂G so that µ η β,q,G = µ β,q,G (· | σ↾ ∂G = η). For a subset ∂G ⊂ V , and FK boundary condition on ∂G is defined as follows: first augment G to a graph G ′ = (V, E ′ ) where E ′ adds edges between any vertices in ∂G not adjacent in G and let E ′ (∂G) be the set of all edges between vertices in ∂G; then an assignment ξ ∈ {0, 1} E ′ is an FK boundary condition on ∂G. Then ξ can be identified with a partition of the vertices of ∂G, where the partition is given by the connected components of ξ. The FK measure with boundary conditions ξ is denoted by π ξ p,q,G and is given by counting k(ω) with connections from ξ in mind.
The red Potts boundary condition is an assignment of σ(i) = 1 to all vertices of ∂G where we always call the first state σ(i) = 1 red, or R. The wired FK boundary condition is that in which all of V (∂G) is in the same boundary component. The free Potts boundary condition corresponds to ∂G = ∅ while the free FK boundary condition corresponds to the partition of ∂G consisting only of singletons {{v} : v ∈ V (∂G)}. The wired/red boundary conditions are ordered as they pick out one of the q ordered phases of the Potts model, whereas the free boundary conditions are disordered as they pick out the disordered phase, where the q states are symmetric.
Edwards-Sokal coupling. The Edwards-Sokal coupling [8] is a coupling of the Potts and FK measures on a graph G that enables us to reduce the study of the q-state Potts model, to the study of the FK model at integer q. The joint probability assigned to (σ, ω), where σ ∈ Ω p is a q-state Potts configuration at inverse-temperature β > 0 and ω ∈ Ω fk is an FK configuration with parameters (p = 1 − e −β , q), is proportional to Planar duality. Throughout this paper we are concerned only with the Potts and FK models on planar graphs, and specifically rectangular subsets Λ n,m ⊂ Z 2 with vertices Λ n,m = 0, n × 0, m := {k ∈ Z : 0 ≤ k ≤ n} × {k ∈ Z : 0 ≤ k ≤ m} and nearest-neighbor edges. For a general subgraph G ⊂ Z 2 , the boundary ∂G will be the set of vertices in G with neighbors in Z 2 − G. When considering rectangles and other graphs where it makes sense, the southern (bottom) boundary of Λ is denoted ∂ s Λ n,m = 0, n × {0} and ∂ e,n,w are analogously defined. Then for rectangles, we observe that ∂Λ n,m = i∈{n,s,e,w} ∂ i Λ n,m .
A very useful tool in the study of these models when G is planar is the planar duality of the FK model. For a planar graph G, let G * be its dual graph. To every FK configuration ω, we associate the dual configuration ω * given by ω * (e * ) = 1 if and only if ω(e) = 0 (where e * is the dual-edge intersecting e). A simple calculation yields that at the self-dual point p sd (q) = √ q 1+ √ q , for G ⊂ Z 2 , we have π G,p sd ,q (ω) = π G * ,p sd ,q (ω * ) .
In the presence of boundary conditions ξ on G whose augmented graph G ′ is planar, the same holds for the corresponding dual boundary conditions ξ * on G * ; these are determined on a case by case basis via the planarity of the augmented graph, but importantly, the wired and free boundary conditions are dual to each other.
Throughout the paper, for two vertices x, y we will write x D ←→ y if they are in the same component in ω↾ D ; when we include an asterisk, we mean x and y are in the same dual-component, i.e. they are in the same component of the dual configuration ω * .
FKG inequality and monotonicity. When q ≥ 1, the FK model satisfies positive correlation (FKG) inequalities: if A and B are increasing events in the edge configuration, for any boundary condition ξ, we have π ξ G,p,q (A ∩ B) ≥ π ξ G,p,q (A)π ξ G,p,q (B). This yields monotonicity in boundary conditions at q ≥ 1: for FK boundary conditions ξ ′ ≥ ξ (ξ is a finer partition than ξ ′ ) and every increasing A, π ξ ′ G,p,q (A) ≥ π ξ G,p,q (A). The domain Markov property. The FK model also satisfies the domain Markov property: for a graph G with boundary conditions ξ, and a subgraph G ′ ⊂ G, where (η, ξ) is the boundary conditions induced on G ′ by connections from η and ξ.
The Potts and FK phase transition. On Z 2 , the FK and Potts models undergo a phase transition from-in the FK setting-existence a.s. of an infinite cluster at p > p c to a.s. no infinite cluster at p < p c . In [1] it was proved that for all q ≥ 1, p c (q) = p sd (q). This corresponds to a Potts phase transition from a unique infinite-volume Gibbs measure when β < β c to q different extremal Gibbs measures corresponding to weak-limits of boundary conditions of the q colors when β > β c .
While for q ≤ 4, the phase transition described above is continuous [7], when q > 4, the phase transition is discontinuous [6]: there are two extremal FK Gibbs measures at p = p c corresponding to the wired and free boundary conditions at infinity, π 1 Z 2 = π 0 Z 2 (resp. at β = β c , q + 1 extremal Potts Gibbs measures corresponding to the q different colors, along with a disordered phase with free boundary conditions at infinity). As a result, we have the following [6,7]: let q > 4 and p = p c ; there exists c(q) > 0 such that Surface tension. The order-disorder surface tension will play a large role in both upper and lower bounds studying the effect of boundary conditions on mixing in the phase coexistence regime. In the low-temperature regime, the metastable phases are the q ordered ones and there is a positive order-order surface tension (see, e.g., the q = 2 case); this leads to sensitivity of mixing times to boundary conditions in Ising/Potts Glauber dynamics (cf., e.g., [17,18]), but not in FK/Swendsen-Wang dynamics (as these are symmetric w.r.t. the q ordered phases). At the critical point, the disordered phase is also metastable and induces similar sensitivity to boundary conditions in FK Glauber and Swendsen-Wang dynamics. Let S n = 0, n × −∞, ∞ and let (1, 0, φ) FK boundary conditions on ∂S n denote those that are wired on ∂S n ∩ {(x, y) ∈ Z 2 : y ≥ x tan φ} and free elsewhere on ∂S n . We will always be taking φ ∈ (−π/2, π/2).
Definition 2.1. The order-disorder surface tension on S n in direction φ is given by (whenever this limit exists) where Z ξ Sn denotes the FK partition function on S n with boundary conditions ξ on ∂S n .

2.2.
Markov chain mixing times. We introduce the relevant dynamical quantities and techniques in the study of mixing times; for an extensive treatment, see [15].
Mixing times. Consider a Markov chain with finite state space Ω, reversible w.r.t. an invariant measure π. Define the total variation distance between measures µ, ν on Ω as where the infimum is over all couplings (X, Y ). Then for a discrete time Markov chain with transition kernel P (·, ·), we define the total variation mixing time of the chain as t mix = inf t : max x∈Ω P t (x, ·) − π tv < 1/(2e) .
Spectral gap and Dirichlet form. One commonly used technique to bound the mixing time of a Markov chain with transition kernel P is to bound the spectral gap of P . A transition matrix P reversible w.r.t., π has largest eigenvalue 1 by Perron-Frobenius, and has real spectrum. Thus we can enumerate its spectrum 1 > λ 1 ≥ λ 2 ≥ ... and define its spectral gap as gap = 1 − λ 1 ; the following relation with t mix is well known: where π min = min x∈Ω π(x). The variational form of the spectral gap is where the Dirichlet form E(f, f ) is given by x,y∈Ω π(x)P (x, y)(f (y) − f (x)) 2 .
Though our main results are stated for the Swendsen-Wang dynamics for the Potts model, as remarked earlier, the proofs all analyze instead the mixing time of the Glauber dynamics for the FK model. We formally define both of these dynamics in the sequel.
Heat-bath Glauber dynamics. Discrete time heat-bath Glauber dynamics for the FK model (FK Glauber dynamics) on a finite graph G with boundary conditions ξ is the Markov chain (X t ) t≥0 defined as follows: at time t it picks an edge e uniformly at random, and resamples ω(e) via π ξ to obtain X t+1 . Though this paper does not consider it, the heat-bath Potts Glauber dynamics (X ′ t ) t≥0 is defined similarly, where at each time step a site v is picked uniformly at random and Swendsen-Wang dynamics. Theorems 1-3 all treat the Swendsen-Wang dynamics [22], which, for the q-state Potts model on a finite graph G = (V, E) at inverse temperature β, is the following discrete-time Markov chain. Given that the Markov chain is at state σ ∈ [q] V at time t, generate state σ ′ ∈ [q] V at time t + 1 as follows.
(1) To σ, assign an (random) FK configuration (ω(e)) e∈E defined as follows: for every edge e = (x, y), independently, set ω(e) to be closed with probability 1 if σ x = σ y and with probability e −β if σ x = σ y . (2) For every connected component of ω, reassign all the vertices in the cluster, collectively, an i.i.d. color in [q], to obtain the new configuration σ ′ . One can check using the Edwards-Sokal coupling of the FK and Potts models, that the Swendsen-Wang dynamics is reversible with respect to µ G,β .
Remark 2.2. The definition of the Swendsen-Wang dynamics indicates why it has fast mixing at low-temperature whereas the Potts Glauber dynamics slows down on (Z/nZ) 2 (see Theorem 4 of [10]). When β > β c , the metastable states are the q ordered phases, corresponding to the q colors; but step 2) above recolors all FK clusters, and in particular, reassigns the large macroscopic cluster of the ordered phase an i.i.d. color, allowing the dynamics to easily jump between the q metastable states. However, as it relies heavily on the FK representation of the model, the order-disorder energy barrier is still hard to overcome when both ordered and disordered phases are metastable.
Comparison between cluster and Glauber dynamics. The following estimates allow us to reduce the analysis of the Swendsen-Wang dynamics to the analysis of FK Glauber dynamics, and demonstrate why our results do not carry over to Glauber dynamics for the Potts model. 23,24]). Let q ≥ 2 be integer. Let gap p and gap fk be the spectral gaps of discrete-time Glauber dynamics for Potts and FK model on a graph on m edges and maximum degree ∆, resp., and let gap sw be the spectral gap of Swendsen-Wang. Then
(2.5) Theorem 2.3 implies, in particular, that the mixing time of the FK Glauber dynamics and Swendsen-Wang dynamics on a graph G are comparable up to polynomial factors in the number of vertices. However, the Potts Glauber dynamics can be much slower than both, e.g., on the n × n torus when β > β c (q).
Potts boundary conditions and mixing. The comparison estimates above are only valid in the absence of boundary conditions, while the focus of this paper is the influence of various fixed boundary conditions. In particular, the estimates of Theorem 2.3 hold immediately for Λ n,n with free or periodic (the torus) boundary conditions. For other boundary conditions, we can deform Λ n,n as in the following remark; this could, however, distort ∆ by an order n factor, leading to an exponential in n cost in (2.4).
Remark 2.4. For any FK boundary condition ξ on G, we can define a (not necessarily planar) graphG by identifying all vertices of every boundary component of ξ with a single vertex inG, and keeping the same edge structure. Then the FK Glauber dynamics onG with free boundary conditions is the same as that on G with boundary conditions ξ. In such a case, let ∂G denote the set of vertices inG that arise from the boundary components of G.
Remark 2.5. The exponential dependence on ∆ in Eq. (2.4) can be improved to exponential in the maximum degree of all but one vertex (see [23, Theorem 1']), whence ∆ in Eq. (2.4) can be replaced with the second largest degree of G. This implies that Theorem 1 in fact also holds for the Potts Glauber dynamics.
The following is a consequence of spin symmetry of the Swendsen-Wang dynamics.
Fact 2.6. Consider Swendsen-Wang dynamics on G with Potts boundary conditions η, by considering the graphG, where boundary vertices of each color are identified as single vertices. Let gap sw,η,G be the spectral gap of Swendsen-Wang dynamics on G with b.c., η, and let gap sw,η,G be the spectral gap of Swendsen-Wang dynamics onG with ∂G assigned the colors given by η. If ∂G consists of at most one vertex, then gap sw,η,G = gap sw,η,G = gap sw,0,G . Remark 2.5 and Fact 2.6 imply that for Λ n,n with FK boundary conditions ξ with at most one nontrivial boundary component, corresponding to Potts b.c., η (an assignment of a color, e.g., red, to the nontrivial component of ξ and no other color assignments), gap fk gap sw n 2 log n · gap fk . Notice that all of the boundary conditions considered in Theorems 1-3 are of this form. Thus, it suffices to prove Theorems 1-3 for the FK Glauber dynamics. From now on, when we write gap with no subscript we mean it to refer to the FK Glauber dynamics.
Monotonicity and the grand coupling. A discrete-time Markov chain with state space Ω and transition kernel P is monotone if µP νP for every two probability distributions µ, ν on Ω such that µ ν; we say that a spin system is monotone whenever single-site Glauber dynamics for it is monotone (cf. [15, §22.3]).
The random mapping representation of the discrete time FK Glauber dynamics on a graph G = (V, E) views the updates as a sequence (J i , U i ) i≥1 , in which J i 's are i.i.d. uniform edges (the updated locations), and the U i 's are i.i.d. uniform on [0, 1]: starting from an initial configuration ω 0 , at time T i , writing J i = (x, y), the dynamics The grand coupling for FK Glauber dynamics is a coupling of the chains from all initial configurations on G which, via the above random mapping representation, uses the same update sequence (J i , U i ) i≥1 for each one of these chains. Using this representation and the FKG inequality, one sees that heat-bath Glauber dynamics for the FK model at q ≥ 1 is monotone: for every two FK configurations ω 1 ≥ ω 2 and every t ≥ 0, we have P t (ω 1 , ·) P t (ω 2 , ·).
Thus, for q ≥ 1, this coupling preserves the partial ordering of the Markov chains started from all possible initial configurations, at all times t ≥ 0. In particular, under the grand coupling, the value of an edge e in Glauber dynamics at time t from an arbitrary initial state ω 0 , is sandwiched between the corresponding values from the free and wired initial states; thus, by a union bound over all edges, Censoring inequalities. The Peres-Winkler censoring inequalities [20] for monotone spin systems allow one to "guide" the dynamics to equilibrium, using that prohibiting updates at various sites for periods of time can only slow down mixing.
. Let µ T be the law of the discrete-time Glauber dynamics at time T for a monotone spin system with state space Ω = S Λ and stationary distribution π, whose initial distribution µ 0 is such that ω → µ 0 (ω)/π(ω) is increasing w.r.t. the partial order on Ω. Set 0 = t 0 < t 1 < . .
be subsets of Λ, and letμ T be the law at time T of the censored dynamics, started at µ 0 , where only updates in Λ i are kept during the times {t i−1 + 1, . . . , t i }. Then µ T − π tv ≤ μ T − π tv and µ T μ T ; moreover, both µ T /π andμ T /π are increasing.
Theorem 2.7 implies in particular that if we consider two chains-the standard FK Glauber dynamics X t ∼ P t (1, ·), and the modified dynamicsX t ∼P t (1, ·) which at certain times sets all bonds in predetermined subsets of G to be wired-thenP t (1, ·) is always farther from π and from P t (0, ·) in total-variation than P t (1, ·) is.

Consequences of positive surface tension
3.1. Cluster expansion. We first introduce the cluster expansion framework used to prove that the FK order-disorder surface tension τ 1,0 (φ) as defined in Definition 2.1 is positive for all large q at p = p c (q). We skip many of the details here as understanding them is not necessary to the equilibrium estimates we require ( §3.2). This approach was extensively developed in [5] for the low-temperature Ising model when β ≫ β c , and then extended to the critical Potts/FK model for large q (at the discontinuous phase transition point) in [19] where one can find more details on the below. Though these cluster expansion techniques only go through at p = p c when q is sufficiently large, all of the interface estimates are expected to hold at p c whenever τ 1,0 (φ) > 0, so in particular, for every q > 4.
In what follows, two (primal) edges e, f are adjacent if they share a vertex. Two primal edges e, f are co-adjacent if e ⋆ is adjacent to f ⋆ . Connectedness and coconnectedness are defined naturally with respect to adjacency and co-adjacency. For an edge subset A ⊂ E(Z 2 ), the (edge)-boundary of A is the set of all edges in A that are co-adjacent to E(Z 2 ) \ A. The co-boundary of A is the set of edges in E(Z 2 ) \ A that are adjacent to A. Definition 3.1. Let D be a connected subgraph of Z 2 and let a, b be two marked boundary points on ∂D. Consider Dobrushin boundary conditions that are wired on the clockwise segment (a, b) and free on (b, a) (where if D is infinite, simply connected, we define these boundary arcs in the natural way). Then for an FK realization ω on D, the primal-FK interface (or simply the FK interface), I = I(ω), is defined as follows: (1) Consider the dual-component of the boundary arc (b, a) in ω, and consider its co-boundary (a set of closed dual-edges in ω).
(2) This co-boundary has a unique co-connected component, call it Γ ⋆ (ω), that is incident to the boundary arc (a, b). The primal-FK interface I is the set of all primal edges that are dual to edges in Γ ⋆ (ω). (Notice that this interface is not necessarily a simple path, but it is connected, and will have no cycles.) Define the dual-FK interface analogously as follows: consider the connected component of open edges touching the boundary arc (a, b) and consider its co-boundary (a set of closed primal edges). This co-boundary has a unique coconnected component that is incident the boundary arc (b, a); the set of dual-edges that are dual to edges in this co-connected component will be the dual-FK interface. For more general boundary conditions, there will be a compatible collection of interfaces between all boundary segments that are wired.
Recall that we defined the infinite strip S n = 0, n × −∞, ∞ . Definition 3.2. For any angle φ ∈ (− π 2 , π 2 ), an edge-cluster weight function Φ(C, I) is any real-valued function with first argument that is a connected set of edges in S n and second argument that is a possible realization of an FK interface with Dobrushin boundary conditions that are wired on the clockwise arc between (0, 0) to (n, ⌊n tan φ⌋), such that for some λ > 0 and every C, I, is the minimum number of edges in a connected subset of E(Z 2 ) that contains all the boundary edges of C. More generally, for an edge-cluster weight function Φ and a domain V n with Dobrushin boundary conditions between (0, 0) and (n, n⌊tan φ⌋), its partition function is given by where the sum runs over all possible FK order-disorder interfaces in V n . There exists a unique edge-cluster weight function Φ o/d -called the FK order-disorder weight function-such that the probability that the FK interface in S n under the (1, 0, φ) boundary condition is I is given by The FK order-disorder weight function Φ o/d is made explicit in [19,Proposition 5].
By adapting the methods of [5] to the FK cluster expansion, the following large deviation estimate on order-disorder interface fluctuations was obtained in [19].
Consider the critical FK model on S n and fix a δ > 0. There exist some q 0 and c(δ) > 0 such that for all φ ∈ [− π 2 + δ, π 2 − δ] and every q ≥ q 0 , every h ≥ 1 and n ≥ 1, The following is a finer result, that reformulates results of [5] in the FK setting. Define the cigar-shaped region U κ,d,φ for every d > 0 and κ > 0 by be the FK order-disorder weight function, and letΦ be any function satisfying Moreover, the order-disorder surface tension τ 1,0 (φ) (given in Def. 2.1) satisfies

4)
and the large deviation estimate of Proposition 3.3 holds for V n and d/2.

3.2.
Surface tension estimates on subsets of Λ n,n . In this section, we extend the surface tension estimate of Proposition 3.3 to tilted half-infinite and finite strips. We first define subsets of S n = 0, n × −∞, ∞ we consider in obtaining the desired sub-exponential upper bounds of Theorems 1-2 (see Figure 2).
Definition 3.5. For every n, which will be understood by context and omitted from the following notation, define a tilted strip Also, define the half-infinite strips, Figure 2. An illustration of some of the sets and interfaces considered in Definition 3.5 and Proposition 3.6. The (1, 0, φ) boundary conditions are depicted with bold lines being wired and dashed lines being free.
The following estimate-extending an estimate of [19], which in turn adapts [5] to the FK model, to half-infinite strips-is a consequence of monotonicity of the FK model and Proposition 3.3.
Proposition 3.6. Fix δ > 0 and let q be sufficiently large. For any b ∈ R and φ ∈ [− π 2 + δ, π 2 − δ], consider the critical FK model on H + b,φ ⊂ S n with boundary conditions (1, 0, b, φ) denoting wired boundary conditions on ∂H + b,φ ∩ H + b+1,φ and free boundary elsewhere on ∂H + b,φ . There exist constants A > 0 and c(δ, q) > 0 such that the order-disorder interface I, satisfies Proof. The proposition was proven in the case φ = 0 in Proposition 4.2 of [10] combining monotonicity of the FK model in boundary conditions with Proposition 3.3. The same proof carries over to the case φ = 0 as long as φ is uniformly bounded away from ± π 2 as the surface tension estimate of Proposition 3.3 on S n is expressed in that setup.
The following is the main equilibrium estimate we will use in the proofs of subexponential mixing for general Dobrushin boundary conditions.
b+1,φ and free elsewhere on ∂(S b,m,φ ∩ Λ n,n ). Then there exists c(δ, q) > 0 such that We need the following preliminary estimate (see Figure 3 as a guide).
Lemma 3.8. Fix δ > 0 and let q be sufficiently large. For any b ∈ R and φ ∈ [− π 2 + δ, π 2 − δ], consider the critical FK model on H + b,φ ∩ Λ n,n with (1, 0, b, φ) boundary conditions denoting wired on ∂(H + b,φ ∩ Λ n,n ) ∩ H + b+1,φ and free elsewhere. There exists c(δ, q) > 0 such that Proof. Let B = H − b+a,φ ∩ H + b,φ ∩ Λ n,n and consider the region S ′ = H + b,φ ∩ H − n,0 . For a general domain D ⊂ S n let the boundary conditions (1, 0, b, φ) on it denote wired on ∂D ∩ H + b+1,φ and free elsewhere on ∂D. By monotonicity in boundary conditions, and then inclusion of events, . The application of the FKG inequality here is valid as under (1, 0, b, φ) boundary conditions, events of the sort "I exceeds some height a", are decreasing events, as adding edges only pushes the order-disorder interface down. This property will be used throughout the paper. By monotonicity in boundary conditions again, By Proposition 3.6, there exists a c(δ, q) > 0 such that Proof of Proposition 3.7. As before, let (1, 0, b, φ) boundary conditions on D ⊂ Z 2 denote those that are wired on ∂D ∩ H + b+1,φ and free on ∂D ∩ H − b+1,φ . By a union bound, write ) and consider the two quantities independently. Observe that the first event on the right-hand side is an increasing event, while the second event is a decreasing event. By reflection symmetry and self-duality, if we prove the desired bound on the latter, for general b, h, φ, it implies the former also. By monotonicity in boundary conditions, The right-hand side above is exactly the probability bounded in Lemma 3.8, from which, along with the symmetry noted above and (3.5), the desired upper bound follows.
We will also need the following bound in order to prove the mixing time upper bounds on cylinders in §4.3. It is an adaptation of the proof of [18,Lemma A.6] from the Ising model to the FK model via Propositions 3.3-3.4, and we omit some details. Proposition 3.9. Fix q to be large enough and ε > 0, and consider the critical FK model on Λ n,h for n 1 2 +ε ≤ h ≤ n with 1, 0 boundary conditions denoting wired on ∂ s Λ n,h and free elsewhere. For ρ ∈ (0, 1) and δ small enough, there exists c(ε) > 0 such that Recall the definition of the cigarshaped region U κ,d,φ in (3.2). Following [18], for every − log 2 n + 2 ≤ i ≤ log 2 n − 2, let z i be the nearest vertex to Let U z i ,z i+1 be the cigar shaped region and φ z i ,z i+1 is the angle of the vector from z i to z i+1 (see, e.g., [18, Fig. 8]). By monotonicity in boundary conditions and {I ∩ B = ∅} being an increasing event, where (1, 0) boundary conditions on ∂H − h,0 are wired on ∂H − h,0 ∩H − 0,0 and free elsewhere. Since U z i ,z i+1 ∩ B = ∅ for all i, if I ⊂ |i|≤log 2 n U i , then the desired property holds.
Lemma A.6 of [18] gives a lower bound on the partition function restricted to interfaces contained in U i as defined above in the setting of the Ising model; the same proof extends that lower bound to the partition function of such FK interfaces, noting that Proposition 3.4 is an analogue of [5,Theorem 4.16]: there exists c > 0 such that where an error of (log n) O(1) was absorbed into the term c(ρh) 2 /n via the assumption that h ≥ n 1/2+ε and a choice of a suitable constant c > 0. Furthermore, Proposition 3.4, in particular (3.4), implies there exists c > 0 so that Then writing the desired probability as in [18, (A.36)] as and plugging in the above lower bound on the numerator and upper bound on the denominator, concludes the proof.

Sub-exponential mixing with symmetry-breaking boundary
In this section we prove sub-exponential upper bounds on the inverse spectral gap of Swendsen-Wang dynamics for general Dobrushin boundary conditions. Here, there is a single high probability minimizer of the surface tension for the model, breaking the order-disorder phase symmetry that induces slow mixing in Section 5. In the proofs in this section, we will need to consider random boundary conditions on Λ n,m . Definition 4.1 (wired/free-at-infinity b.c.). Define the wired-at-infinity boundary conditions as the distribution over boundary conditions on ∆ ⊂ ∂Λ n,m given by the distribution on partitions of ∆ induced by π 1 Z 2 (·↾ Z 2 −Λn,m ), so that in general, the connections interior to Λ n,m do not count towards the induced boundary conditions. Define the freeat-infinity boundary conditions analogously as the distribution given by π 0 Z 2 (·↾ Z 2 −Λn,m ). We say that a distribution P on boundary conditions on ∆ dominates P ′ if ξ ′ ∼ P ′ is a stochastically finer partition than ξ ∼ P. Moreover, if boundary conditions are given by different distributions on different subsets of ∂Λ n,m , then define the overall distribution on boundary conditions by sampling the boundary partitions independently on each of the boundary subsets (with no connections between the subsets). We say such a boundary condition piecewise dominates wired-at-infinity and is dominated by free-at-infinity.
A particular example of random boundary conditions that we will be useful later on is the following: if R ⊂ D are two concentric rectangles, then the boundary conditions induced by the connections from ω↾ D−R , where ω is sampled from π 1 R , dominate the wired-at-infinity boundary conditions. 4.1. A canonical paths estimate for 2D random cluster models. The canonical paths technique has yielded (see [3,4,13,21]) a very useful upper bound on the mixing time of spin-systems on general graphs; namely, the mixing time is at most exponential in the cut-width of the underlying graph (so that on rectangular subsets of Z 2 , it is at most exponential in the shorter side length). In the case of the random cluster model, the long-range interactions complicate this for general boundary condition, but we prove a modified version of such an estimate for a wide class of boundary conditions. Proposition 4.2. Let q > 4, p = p c (q), and φ ∈ (− π 2 , π 2 ). Consider the FK model on S = S b,h,φ ∩Λ n,n with boundary conditions ξ ∼ P, where P is arbitrary on ∂S ∩∂ e,w Λ n,n , and piecewise dominates wired-at-infinity or is dominated by free-at-infinity on each of the other sides of ∂S. There exists C q > 0 such that for every sequence f n → ∞, the FK Glauber dynamics have . The proof will use the following straightforward comparison estimate.
wherek(ξ, F ) is the number of components of ξ that intersect both ∂F and ∂S − ∂F .
Proof. By the domain Markov property, the difference between π ξ S (· | ω↾ F = ζ) and π ξ S (· | ω↾ F = ζ ′ ) is in the boundary conditions that ζ ∪ξ and ζ ′ ∪ξ induce on S −F . The boundary partitions induced can differ arbitrarily on ∂F − ∂S contributing a factor of q |∂F −∂S| ; on the other hand, since both configurations use ξ boundary conditions on S, the boundary partitions on the rest of ∂(S − F ) can only differ if ζ induces additional boundary connections between distinct boundary components of ξ that reach ∂S − ∂F ; this accounts for the factor of qk (ξ,F )−1 .
Proof of Proposition 4.2. We modify the proof of the canonical path estimate for spin-systems with short-range interactions to the present setup. Partition every segment L of ∂S \ ∂ e,w Λ n,n on which ξ is independently sampled, into ⌊ |L| fn ⌋ sub-segments ℓ i of f n vertices each, and possibly an additional exceptional segment ℓ 0 . For every edge e ∈ L, let Γ e be the set of components of ξ that contain vertices on both sides of L − {e}, which we hereafter refer to as bridges. Let (4.1) We first prove that for every ξ ∈ f n , the FK Glauber dynamics on S has This will follow from a standard application of the canonical paths argument for spinsystems with short-range interactions. Namely, label the edges in S lexicographically in their midpoint, first by horizontal coordinate, then by vertical coordinate, and define the path γ ω,ω ′ (identified with a sequence of edges in Ω fk between FK configurations, ω and ω ′ ) as follows: let e l 1 , ..., e l k be the sequence of edges on which ω(ẽ l i ) = ω ′ (ẽ l i ), labeled in their lexicographic ordering. The i'th edge in γ ω,ω ′ will then be between configuration η = ω ′ ↾ {e 1 ,...,e l i −1 } ∪ ω↾ {e l i ,...,e |E(S)| } and its neighbor η ′ which also has η ′ (ẽ i ) = ω ′ (ẽ i ). Also, let η * be the configuration that is instead given by η * = ω↾ as |∂F − ∂S| ≤ 2h andk(ξ, F ) ≤ 2h + f n . This follows from the fact that ξ ∈ E fn and the nested structure of boundary bridges, and the fact that the sides with arbitrary boundary conditions have height at most 2h. By construction, for every transition (η, η ′ ), the map (ω, ω ′ ) → (η, η * ) is injective. Moreover, the probability of making any transition in Ω fk is bounded below by 1 1+ √ q . Putting all this together, by the path method we see that for every ξ ∈ E fn , Eq. (4.2) holds and by (2.2), the corresponding bound with an extra factor of O(|E(S)|) also holds for the mixing time.
It remains to bound the P-probability that ξ ∈ E fn . Fix a segment L of ∂S \∂ e,w Λ n,n on which P is piecewise sampled, and fix a sub-segment ℓ i , then take a union bound over all such segments and all ℓ i . Moreover, suppose that the boundary conditions on the segment L are dominated by free-at-infinity (the estimate for the case when the distribution of P on ξ↾ L dominates wired-at-infinity follows by similar reasoning).
Since |Γ e | ≥ 1 is an increasing event, it suffices to show π 0 Z 2 (∃e ∈ ℓ i s.t. |Γ e | = 0) ≥ 1 − O(n 2 e −Cqfn ) . However, by planarity of boundary conditions induced by π 0 Z 2 on ∂S, the complement of the left-hand side is the event that there exist x, y in the two parts of L − ℓ i such that x ←→ y, which, if C q is the constant from (2.1), has probability at most (For the wired-at-infinity boundary conditions, observe that in order for |Γ e | > 1 for every e ∈ ℓ i , there must exist x, y in the two parts of L − ℓ i such that x * ←→ y: this is in turn a decreasing event with exponentially decaying probability under π 1 Z 2 .) Taking a union bound over at most 4n sub-segments ℓ i of various segments L, we obtain that

Dobrushin boundary conditions.
In this section, we consider the mixing time of Swendsen-Wang dynamics with boundary conditions that are free on a subset of ∂Λ and red elsewhere. While §5 demonstrates that such boundary conditions can induce a slow mixing (at least exp(cn)) by respecting the order-disorder phase symmetry, this section will establish that the mixing time is faster (at most exp(n 1/2+o(1) )) under boundary conditions that have a single order-disorder interface.
Define a general class of order-disorder Dobrushin boundary conditions, whose FK representation is wired on one connected boundary arc and free elsewhere. Let a n , b n be two distinct points on ∂Λ n,n . For marked boundary points (a n , b n ), FK Dobrushin boundary conditions are those that are wired on the clockwise (starting from the origin) arc (a n , b n ) and free on (b n , a n ).
Sketch of proof. Our proof of Theorem 2 adapts the proof of t mix exp(c √ n log n) in [17] for the low-temperature Ising model with plus boundary conditions, but using the censoring inequality Theorem 2.7 instead of the block dynamics approach of [17]. For Dobrushin boundary conditions between (a, b), we sequentially (at times t i ) censor all updates except those in the strip B i of height √ n log n parallel to the line segment a, b such that B i and B i+1 overlap on half their height. On the one hand, the canonical paths estimate Proposition 4.2 bounds the mixing time of B i by exponential in its height, so that we take t i −t i−1 = O(e c √ n log n ). On the other hand, by Theorem 2.7, if the censored chains started from all wired and all free are coupled with high probability, that bounds the mixing time of the original chain. To couple these two chains, we systematically push the interfaces of the chains started from these initial configurations down (resp., up), until they are within O( √ n log n) of each other. This is possible because with high probability (see Lemma 3.8) the interface of the bottom boundary of B i (where the free initial configuration is seen) never reaches the top of B i−1 and the censored dynamics continues pushing the interface down to O( √ n log n) distance of a, b (see Figure 5).
Proof of Theorem 2. If a n , b n are on the same side of ∂Λ n,n , rotate Λ n,n so that they are both on ∂ s Λ n,n and the angle of the interface between them will be zero. Then the proof below when a n and b n are on different sides applies identically; the only difference is the boundary conditions in the last step of recursion, where the identity coupling still couples all FK chains with probability 1 in the mixing time of that last block. Now suppose a n = (a 1 n , a 2 n ) and b n = (b 1 n , b 2 n ) are on different sides of ∂Λ n,n ; by rotational and reflective symmetry and self-duality, we can take a n ∈ ∂ w Λ n,n to be the first point encountered clockwise from the origin, and ensure that φ n = tan −1 ( a 2 . Fix any such choice of a n , b n ∈ ∂Λ n,n and let φ = φ n . We establish the theorem for FK Glauber dynamics with (a n , b n ) Dobrushin boundary conditions. Throughout this proof, let c 1 = c 1 (q) > 0 be a large enough constant (e.g., 5/C q for C q from Proposition 4.2 would suffice). Define the overlapping blocks of the B j may be empty, and henceforth if B j is empty, we say any associated events hold trivially. Let for K to be chosen large later. Define the censored chainX t : between times t i−1 and t i censor all updates except those in B i ∪ B −i . LetX 1 t be the censored chain started from X 0 = 1 andX 0 t be the censored chain started fromX 0 = 0. By Theorem 2.7 and (2.6), it suffices to show that there exists a coupling ofX 0 t andX 1 t such that since for any K, c 1 fixed, we have t N exp(O( √ n log n)) as desired.
We now define a monotone coupling ofX 1 t andX 0 t which satisfies the above. For each i = 1, . . . , N , define the event We can then write under our coupling, . The bounds on P(A + N −1 ) and P(A − N −1 ) are analogous (using the duality of the FK model) and therefore we only bound the former. Abusing notation slightly, when we consider the restriction of the chainX 1/0 t to a boundary ∂S , we mean the boundary conditions induced on that line byX 1/0 t ↾ Λ−S . We will prove the following inductively. Claim 4.5. There exists c(q) > 3 so that, for large enough K, c 1 , the following holds.  Figure 5. The green and blue blocks B k , B k+1 are updated by the censored dynamics in two consecutive steps; with high probability, the chain (X 0 t k ) k pushes its interface down toward a, b by ℓ at every step, and is subsequently coupled to (X 1 t k ) k on the growing gray region.
(2) For every 1 ≤ i ≤ N − 1, Proof of Claim 4.5. For the base case, (1) holds trivially because the outer boundary on B 1 is just given by the boundary of ∂Λ which will be all-wired there. The base case proof of (2) is just a simplification of the proof of the inductive step for (2) so we do not repeat it here. Now suppose both (1) and (2) hold for some i − 1 and prove they hold for i for a c(q) > 0 we will pick later. By the inductive hypothesis, with probability , and the boundary conditions on ∂ n B i dominate wired-at-infinity. In particular, since on the other sides of B i both chains' boundary conditions are either all-free or all-wired, by Proposition 4.2, with probability 1 − O((i − 1)n −c ) − O(n −Cqc 1 +2 ) (where C q > 0 is the constant from that proposition) the mixing time of FK Glauber dynamics with boundary conditions given byX 1 t i ↾ ∂B i is at most |E(B i ∪ B −i )| 2 e 4(4ℓ+c 1 log n) log q . Now suppose that both of these events hold and consider the probability thatX 1 By submultiplicativity of total-variation distance, with probability 1 − O(n −K ) there exists a coupling ofX . It remains to compute the probability of (1) obtaining boundary conditions under πX on ∂ n B i+1 , and (2) succeeding in coupling πX (and therefore on all of R + i ).
Under the above events, let (ζ, 0) be the boundary conditions induced on ∂B i byX 0 and (ζ, 1) be those induced byX 1 t i−1 where ζ is a random boundary condition sampled from a distribution dominating π 1 Then, the monotone coupling of π ζ,0 B i to π ζ,1 B i couples the two on B i − B i+1 whenever the bottom-most horizontal crossing I in the sample from π ζ,0 B i has I ⊂ B i+1 (see Figure 5). In that case, by revealing dual-edges from the bottom up, the configurations from π ζ,0 B i and π ζ,1 B i could be coupled above that interface and in particular on all of B i − B i+1 . (Observe that conditioning on the configuration below the interface, in order to reveal I, cannot affect the boundary conditions above it because on each side of ∂(R + i ∪ B i+1 ) the boundary conditions are all-wired or all-free and additional connections cannot be induced (cf. the boundary bridges of [9]).) Observe, also, that when considering ζ dominating the boundary conditions induced by π 1 , since the boundary on ∂B i+1 ∩ ∂Λ is wired, by the domain Markov property, the boundary conditions induced byX 0 t i+1 on ∂ n B i+1 when I ⊂ B i+1 holds will dominate π 1 ). This implies part (1) if we can bound π ζ,0 ∂nB i (I ⊂ B i+1 ). Thus, for both (1) and (2) of the induction, it only remains to bound the probability where (1, 0) boundary conditions on R + i ∪ B i are free on ∂ s B i and wired elsewhere. In that case, Lemma 3.8 (noting that the estimate there was independent of b) with the choice of a = 1 2 c 3 √ n log n implies that the probability in (4.4) is at most O(n −6 ). Combining all of the above, the probability that items (1) and (2) hold is at least which concludes the proof of the induction as long as we take c 1 , K large enough that that the latter three terms are all o(n −c ).  (1), the boundary conditions they induce dominates wired-at-infinity on ∂ n B N with probability at least 1 − o(n −2 ) and likewise, are dominated by free-at-infinity on ∂ s B N with similar probability. Therefore, by time t N , by submultiplicativity of total-variation distance and Proposition 4.2, we have and for c 1 , K large enough, the right-hand side is 1 − o(n −2 ).

4.3.
Sub-exponential mixing on cylinders. For the rectangle Λ n,n define boundary conditions (p, R) (resp. (p, 0) or (p, R, 0) boundary conditions) to be periodic boundary conditions on ∂ n,s Λ n,n and red boundary conditions on ∂ e,w Λ n,n (resp. free on ∂ e,w Λ n,n or red on ∂ w Λ n,n and free on ∂ e Λ n,n ). We prove the mixing time upper bounds on cylinders with the above boundary conditions (Theorem 3) at the same time. In what follows, we use c > 0 to denote the existence of a constant (possibly depending on q), where different appearances of c at different places may refer to different values. The proof builds on the proof of Theorem 2 in that we use the censoring inequalities to push the FK order-disorder across Λ n,n in order to couple the chainsX 1 t andX 0 t . We consider the censored dynamics that sequentially update N = O(n 1 2 −ε ) overlapping vertical strips of width n 1 2 +ε , ordered from left to right. However, unlike the case in Theorem 2, our strips do not have wired boundary conditions on three sides (their boundary conditions on the top and bottom are periodic), and therefore, the interface is pushed to the next strip to be updated with probability exp(−cn 2ε ) (rather than 1 − o(1)): see Figure 6. Thus, with probability exp(−cn 1 2 +ε ) the interface moves to next strip in N consecutive time steps, so that one will succeed, with high probability, at pushing the interface completely across Λ n,n after exp(n Since the boundary conditions on ∂ n,s Λ n,n are periodic, each B 2i can be viewed as a single connected rectangle with boundary ∂ n,s B 2i = (i − 1)ℓ, (i + 1)ℓ × {⌊ n 2 ⌋ + δn, ⌊ n 2 ⌋ − δn} . We prove the mixing time upper bound for (p, 1, 0) boundary conditions. We will pause to comment where the (p, 1) boundary conditions would behave differently (namely only when updating block B 2N +1 ), and on why this does not affect the proof. The (p, 0) boundary conditions can be treated by the dual version of the argument we present.
We will cycle through the blocks B i periodically, so define Define the following censored Markov chainX 1 t (resp.,X 0 t ) started from initial configuration 1 (resp., 0): for all i ≥ 0, Let f n = n by Theorem 2.7 and (2.6), it will suffice to show that We begin with a uniform upper bound on the mixing times of B i .
Claim 4.6. Let m ≤ (2N + 1) exp(n 1 2 +2ε ) and, for every i ≤ m, define the event where the superscriptX ω 0 t i−1 denotes boundary conditions induced byX ω 0 t i−1 on B i . Then where C q > 0 is the constant given by Proposition 4.2.
Proof of Claim 4.6. Let Ξ i be the event that the law of the boundary conditions on B i underX 0 t i−1 piecewise dominate/are dominated by wired/free-at-infinity, resp., and likewise forX 1 t i−1 . We prove inductively that for every m ≤ (2N + 1) exp(n 1 2 +2ε ), The base case, m = 1, has boundary conditions that are wired on ∂ w B 1 and free/wired on ∂ n,s,e B 1 , and thus Υ 1 ∩ Ξ 1 holds with probability 1 by a canonical paths estimate (there are no distinct bridges). Suppose now that (4.8) holds for some m; to show that it holds for m + 1, it suffices to show that the boundary conditions induced byX 0 tm , i.e., the chainX 0 t (the bound for the chainX 1 t follows symmetrically). Assume that the event i≤m (Υ i ∩ Ξ i ) holds. First of all, we notice that for any i satisfying Υ i , by the sub-multiplicativity of total-variation distance and definition of t i , thus, a union bound over all i ≤ m = O(N exp(n 1 2 +2ε )) implies that we may construct a coupling of (X 0 t ) and some random variablesZ 1 , . . . ,Z m such that We now claim that the boundary conditions induced byX 0 tm on B m+1 are such that they piecewise dominate/are dominated by wired/free-at-infinity on ∂B m+1 . Consider the case where m is even (the case m is odd follows analogously). According to Ξ m−1 , if we denote by ζ the boundary conditions induced byX 0 t m−2 on ∂B m−1 , then ζ piecewise dominates/is dominated by wired/free-at-infinity. Hence, when sampling from π ζ B m−1 , there would be a well-defined FK order-disorder interface I between the boundary subsets that are alternately wired and free. For every such interface, by the domain Markov property, the marginal under π ζ B m−1 on each of the connected components of E(B m ) \ I either dominates wired-at-infinity or is dominated by free-at-infinity.
As a consequence, the boundary conditions on ∂ n,s,w B m are piecewise sampled from distributions dominating/dominated by wired/free-at-infinity, as are the boundary conditions on the vertical bisector of (B m−1 \ B m ), a subset of ∂ w B m+1 . Repeating this 24 REZA GHEISSARI AND EYAL LUBETZKY reasoning for the update on B m , we see that the boundary conditions on ∂ n,s,w B m+1 are all sampled from distributions that piecewise dominate/are dominated by wired/free-atinfinity. Finally, the same is true of ∂ e B m+1 as it is either completely free if m ≤ 2N +1 or it is similarly sampled from πX 0 t j−1 B j for some j < m which likewise satisfied Ξ j . Thus we deduce that, except with probability e −n/2 , Ξ m+1 holds. Then by Proposition 4.2, the boundary condition on B m+1 induced byX 0 tm is such that . A union bound over the above errors concludes the proof.
Henceforth, we suppose that the event Υ i holds for all i ≤ (2N + 1) exp(n 1 2 +2ε ), which is the case with probability 1 − exp(−Ω(n 1 2 +3ε )) since f n = n 1 2 +3ε . Note that every time increment of t 2N +1 we make an independent attempt at couplinḡ X 1 t toX 0 t , albeit with initial configurations induced by the chains at the end of the last sweep, and once the two chains are coupled on all of Λ, they will remain coupled for all subsequent times. We will show that there exists c(δ, q) > 0 such that for every k and every two configurations where, in analogy to the proof of Theorem 2, if R i = j≤i mod (2N +1) B j , Eq. (4.10) is sufficient because the probability of not couplingX 0 T andX 1 T by time T (as defined in (4.5)) would then be bounded by In order to lower bound the probability in (4.10), we will construct a monotone coupling of the two chains; therefore it suffices to consider the wired and free initial configurations; by the Markov property, it also suffices to only consider the first sweep k = 1.
Recall from (4.1) that for any rectangle E fn is the set of boundary conditions on B i such that in every boundary segment of length f n , there is an edge with at most one boundary component containing vertices on both sides of that edge (bridge).
is in E fn and dominates the wired-at-infinity distribution on boundary conditions on ∂ w (B 2i+1 ∪ B 2i+2 ).
(2) For every 1 ≤ i ≤ N , the above defined event F i satisfies Proof of Claim 4.7. For the base case of (1), observe that the boundary conditions on ∂ w (B 1 ∪ B 2 ) dominate wired-at-infinity as they are all-wired; the base case of (2) follows as the proof of the inductive step does, so we do not repeat it here. Now assume that items (1) and (2) hold for i − 1 (for c(q) > 0 to be determined later) and show that they hold for i for the same choice of c. Consider the middle rectangle D i = (i − 1)ℓ, (i + 1 2 )ℓ × ⌊ n 2 ⌋ − δn, ⌊ n 2 ⌋ + δn , and let I be the interface bounding the cluster(s) ofX 0 t j ↾ ∂wB j . Define the events whereΓ 2i−1 is the event that I is connected to e 1 , e 2 in B 2i−1 − D i , where e 1 , e 2 are a pair of edges in either side of ∂ w B 2i−1 − ∂ w D i , with at most one bridge over them (such a pair of edges exist since by assumption (1), the boundary conditions on B 2i−1 are in E fn ). It is clear that both Γ 2i−1 and Γ 2i are increasing events. Then, we can lower bound By the inductive hypothesis, P(A c 2i−2 , F i−1 ) exp(−c(i − 1)n 2ε ) and from now on work in the probability space conditioned on are coupled and dominate wired-at-infinity (let ζ denote that random boundary condition). In the following time increment [t 2i−2 , t 2i−1 ), only updates on B 2i−1 are permitted; since we are working on the event Υ 2i−1 , we can coupleX 0 t 2i−1 ↾ B 2i−1 to π ζ,0 B 2i−1 andX 1 t 2i−1 ↾ B 2i−1 to π ζ,1 B 2i−1 with probability 1 − O(e −n ) by (4.9).
Thus, by monotonicity, we consider the probability of the event I ∩ D i = ∅ in Γ 2i−1 holding for a sample from π ζ,0 B 2i−1 . We claim that for some c(δ, q) > 0, where the expectation is over all boundary conditions ξ induced by π 1 Z 2 on ∂ w B 2i−1 and (1, 0) boundary conditions denote wired on ∂ s Λ 1,0 (1−2δ)n,3ℓ . Indeed, the second inequality follows from considering the ℓ-enlargement E ℓ,2i−1 of B 2i−1 which is its concentric rectangle with extra side length ℓ. If there is a wired circuit in E ℓ,2i−1 − B 2i−1 under π 1 Z 2 (by (2.1) this has probability 1 − e −cℓ ), we can replace the expectation over b.c. induced by π 1 Z 2 with an expectation over b.c. induced by π 1 E ℓ,2i−1 . Then extending the free boundary conditions on the other three sides of B 2i−1 all the way to ∂ w E ℓ,2i−1 and rotating yields the second inequality. By Proposition 3.9 with the choices h = 3ℓ and ρ = 5 6 , there exists c(δ, q) > 0 so that the probability in the right-hand side above is at least order e −cn 2ε (see e.g., [10, §5 and Fig. 7] for a similar monotonicity argument). Moreover, by the exponential decay of dual-connectivities, it is clear that for any ζ ∈ E fn , we have π ζ,0 B 2i−1 (Γ 2i−1 ) ≥ η for some η(q) > 0. Thus by the FKG inequality, Let I 2i−1 be the interface revealed by the component of ∂ w B 2i−1 . Observe that because Γ 2i−1 is an increasing event, conditioned on Γ 2i−1 , if we reveal I 2i−1 from east to west, under the monotone coupling of π ζ,1 B 2i−1 to π ζ,0 B 2i−1 the same edges would also be open under π ζ,1 B 2i−1 ; the same is also true of the edges that constituteΓ 2i−1 . Having revealed these sets of open edges under both π ζ,1 B 2i−1 and π ζ,0 B 2i−1 , by the domain Markov property (there can not be distinct bridges over the interface we have revealed), )e −cn 2ε . Now consider the next time increment [t 2i−1 , t 2i ) on B 2i . Under the above events, the configurationX 0 t 2i−1 ↾ D i π 1 Z 2 (·↾ D i ), whence by (2.1), with probability at least 1 − e −cδn , there is a pair of primal horizontal crossings of the top and bottom halves of D i connecting ∂ w B 2i−1 to I 2i−1 . In that case, the distribution on boundary conditions induced byX 0 t 2i−1 (as well asX 1 t 2i−1 ) on ∂ n,s B 2i ∩D i dominates wired-at-infinity. Again, since we are working under the event Υ 2i , we just consider the event in Γ 2i under π ζ,0 B 2i . By applying (2.1) and enlarging the domains under consideration as in the earlier bound on Γ 2i−1 , we obtain for some c(δ, q) > 0, . Here, the boundary conditions (ξ, 0) in the first line denote ξ (over which we take an expectation) induced on ∂B 2i ∩ {(x, y) : x = (i + 1 2 )ℓ}, and free elsewhere on ∂B 2i , and the boundary conditions in the second and third lines denote free on ∂ n of the boundary and respectively ξ and wired elsewhere. The second inequality is a simple consequence of monotonicity in boundary conditions and the third inequality follows from enlarging B 2i by ℓ/2 up to an error of e −cℓ coming from (2.1). By Lemma 3.8 with φ = 0 and b = 1, there exists c(δ, q) > 0 such that the probability on the right-hand side above is bounded below by (1 − O(e −n ))(1 − e −cn 2ε ). In that case, revealing the interface from east to west, we can coupleX 0 t 2i toX 1 t 2i beyond the interface (see also Fig. 6), so Finally, we claim that under the intersection of all the above events, with probability 1 − O(e −cℓ ) − O(n 2 e −Cqfn ), the boundary conditions induced byX 1/0 t 2i on ∂ w B 2i+1 and ∂ w B 2i+2 are in E fn and dominate wired-at-infinity, which combined with (4.12) defines the desired set set F i such that . Recall that the configuration on D i under Γ 2i−1 and F i−1 dominates π 1 Z 2 ↾ D i . Then, with probability 1 − 2e −cδn 1 2 +ε , D i contains two horizontal crossings connecting ∂ w D i to I 2i−1 ; since we are also conditioning on Γ 2i , averaging over configurations on D i , with probability 1 − 3e −cδn 1 2 +ε , ∂ w B 2i+2 is surrounded by a wired circuit inX 0 t 2i . Similarly, underX 0 t 2i , conditional on Γ 2i−1 , Γ 2i and F i−1 , the configuration on D ′ i = iℓ, (i + 2)ℓ × 0, δn ∪ (1 − δ)n, n below the interface revealed by Γ 2i dominates π 1 Z 2 so that with probability 1−2e −cδn 1 2 +ε , there are horizontal primal connections to that interface in both halves of D ′ i . In that case, averaging over configurations in D ′ i with probability 1 − 3e −cδn 1 2 +ε , there is a wired circuit around ∂ w B 2i+1 inX 0 t 2i so that the distribution over boundary conditions induced on ∂ w B 2i+1 also dominates wired-at-infinity. Moreover, as seen in the proof of Proposition 4.2, a boundary condition dominating wired-at-infinity is in E fn with probability 1 − O(n 2 e −Cqfn ). A union bound over the above concludes the proof.
As a result, by item (2) of Claim 4.7, there exists some c(δ, q) > 0 for which P(A c 2N ) exp − cN n 2ε exp − cn 1 2 +ε . Moreover, on that event, with high probability, the boundary conditions on ∂ w B 2N +1 induced by bothX 1 2N andX 0 2N dominate wired-at-infinity. On the event Υ 2N +1 , with probability 1 − O(e −n ) one can coupleX 1 t 2N+1 andX 0 t 2N+1 to agree on B 2N +1 , leading the two chains to be coupled on all of Λ. (It is only at this final step where there is a difference between the (p, 1, 0) and (p, 1) boundary conditions; clearly, if the coupling on B 2N +1 succeeds in the former situation, it also succeeds in the latter.)

Slow mixing with phase-symmetric boundary conditions
For a reversible chain with transition kernel P (x, y) and stationary distribution π, define the edge measure Q between A, B ⊂ Ω and conductance of the chain, Φ, by The Cheeger inequality relates these to the gap (see, e.g., [15, §7]), by stating that The torus. In Theorem 2 of [10], the authors used the above to construct an exponential bottleneck relying heavily on the topology of the torus and the exponential decay of correlations under π 0 Z 2 at a discontinuous phase transition point. We restate the result for the critical Swendsen-Wang dynamics for all q > 4, which follows from the sharp identification in [6] of the discontinuity of the phase transition for all q > 4.
Observe that exploiting the topology of the torus, unlike the other results in this paper, the above requires neither validity of the cluster expansion nor positivity of surface tension, and therefore holds up through q > 4. On the other hand, for q that is sufficiently large, slow mixing at β = β c was previously shown in [2] (in any dimension).
Order-disorder mixed boundary conditions. Although the proof of slow mixing on the torus at a discontinuous phase transition relies heavily on the topology of the torus (see proof of Theorem 2 in [10]) we can-at least for sufficiently large q-use a similar approach to prove slow mixing in the presence of mixed wired-free boundary conditions. Exploiting the self-duality, we see that such boundary conditions still exhibit an exponential bottleneck, slowing down the Swendsen-Wang dynamics.
Definition 5.2. Let a n , b n , c n , d n ∈ ∂Λ n,n be a set of marked vertices ordered clockwise from the origin around ∂Λ n,n (by rotational symmetry, without loss of generality assume a n ∈ ∂ w Λ n,n ). The mixed boundary conditions on (a n , b n , c n , d n ) are those that are red on the clockwise boundary arcs (a n , b n ) and (c n , d n ) and free on (b n , c n ) and (d n , a n )-all connected subsets of ∂Λ n,n . We say that (a n , b n , c n , d n ) are ε-separated if a n ∈ ∂ w Λ n,n , at least one of {b n , c n , d n } is not contained in 0, εn × 0, n , and min i,j∈a,b,c,d; i =j i n − j n ∞ ≥ εn . Remark 5.3. The requirement of ε-separation in our consideration of mixed boundary conditions arises from the fact that if (a n , b n , c n , d n ) were all on ∂ w Λ n,n repeating the proof of Theorem 2 with such boundary conditions would yield that the mixing time is in fact sub-exponential. Clearly, if the four marked vertices are sufficiently close to being on one side or to each other, a similar picture would emerge. The requirement of macroscopic separation ensures that the bottleneck is exponential in n.
With Definition 5.2 in hand, to prove Theorem 1, by rotational symmetry, we wish to prove the following: let q be large, ε > 0, and consider the Swendsen-Wang dynamics for the critical Potts model on Λ = Λ n,n with mixed boundary on (a n , b n , c n , d n ) that are ε-separated. Then there exists c(ε, q) > 0 such that t mix exp(cn) .
Proof of Theorem 1. By (2.5) and Fact 2.6, it suffices to prove the bound for the FK Glauber dynamics with mixed FK boundary conditions that are wired on the boundary arcs (a n , b n ) and (c n , d n ) ⊂ ∂Λ n,n (and the two boundary arcs are wired together) and free elsewhere (denoted by π mixed Λ ). Observe that by planarity, (a n , b n ) ←→ (c n , d n ) = (b n , c n ) * ←→ (d n , a n ) c , and therefore, either π mixed Λ (a n , b n ) ←→ (c n , d n ) ≤ 1 2 , or π mixed Λ (b n , c n ) * ←→ (d n , a n ) ≤ 1 2 .
By self-duality of the class of ε-separated, mixed boundary conditions, we can suppose without loss of generality that we are in the former case.