Olle Haggstrom, Johan Jonasson, Russell Lyons
An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity.
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We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-c…
This is a short account of connections between the Tutte polynomial and the Ising, Potts, and random-cluster models. The four principal elements are the Ising model of 1925, the Tutte polynomial of 1947, the Potts model of 1952, and the random-cluster model of 1972.
Three-point connectivity constant for $q$-state Potts spin clusters
本文为平面临界$q$-态Potts自旋簇引入并显式计算连续统三点连通常数，与虚DOZZ公式相差一个几何常数，并回答了Delfino等人提出的问题。
We prove that for q>=1, there exists r(q)<1 such that for p>r(q), the number of points in large boxes which belongs to the infinite cluster has a normal central limit behaviour under the random cluster measure phi_{p,q} on Z^d, d>=2. Particularly, we can take r(q)=p_g^* for d=2, which is commonly conjectured to be equal to p_c. These results are used to prove a q-dimensional central limit theorems relative to the fluctuation of the empirical measures for the ground Gibbs measures of the q-state Potts model at very low temperature and the Gibbs measures which reside in the convex hull of them. …
We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous comp…
In this article we create a new algorithm for the perfect simulation of the infinite Potts model at a sufficiently small or at a sufficiently high temperature, in particular under the transition phase temperature. We study the model for free boundary conditions and we give some consequences for the constant boundary conditions.
For a family of random-cluster models with cluster weights $q\geq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $\tfrac{1}{q}χ(β)^{2}βJ_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.
Conformal invariance of the planar four-state Potts model and its random-cluster representation
本文证明临界随机簇模型（q=4）的环在标度极限下收敛到CLE(4)，并利用Edwards-Sokal耦合推导出四态Potts模型的共形不变性。
We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establish that for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal $Θ(n\log n)$ steps on $n$-vertex random graphs having a prescribed degree sequence with bounded average branching $γ$ throughout the full high-temperature uniqueness regime $p<p_u(q,γ)$. The family of random graph mode…
We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among $q$ "visible" colours along with the presence of $r$ non-interacting "invisible" colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any $q>0$, as long as $r$ is large enough. When $q>1$, the low-temperature regime displays a $q$-fold symmetry breaking. The proof involves a Pirogov-Sinai analysis applied to this random-cluster representation of the model.
We prove sharpness of the phase transition for the random-cluster model with $q \geq 1$ on graphs of the form $\mathcal{S} := \mathcal{G} \times S$, where $\mathcal{G}$ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q \geq 1$, there exists some parameter $p_c = p_c(\mathcal{S}, q)$, below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts m…
Generalized cluster algorithms for Potts lattice gauge theory
将 Swendsen-Wang 和 invaded-cluster 算法推广到 Potts 格点规范理论，通过 plaquette 随机簇模型实现，在临界点利用同调渗流停止条件，在四维环面上显著加速自相关衰减。
We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $β<β_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the…
Mean-field Potts and random-cluster dynamics from high-entropy initializations
针对高维多模态分布下马尔可夫链最坏情况初始化混合慢的问题，研究完全图上随机簇与Potts模型三类动力学的高熵初始化收敛性，刻画快速混合的乘积测度初始化集并给出基于一维过程近似的证明。
A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $ϕ_{p,q}$, and then a spin ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of Häggström [Ann. Appl. Probab. 9 (1999) 1149--1159] and Häggström and Schramm [S…
We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $Δ\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $Δ$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic …
This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state …
For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on finite tori $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $β\geq 0$. This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen--Wang dynamics mix slowly at the critical temperature. We also provide an efficient algo…
Potts and random cluster measures on locally regular-tree-like graphs
该研究针对局部收敛于d-正则树的一致稀疏有限图，证明Potts自由能密度的Bethe预测收敛性可推出Potts测度与随机团测度的局部弱收敛，并刻画了极限态选择、临界线混合态及零场下的纯态分解。
Correlation inequalities are presented for ferromagnetic Potts models with external field, using the random-cluster representation of Fortuin and Kasteleyn, together with the FKG inequality. These results extend and simplify earlier inequalities of Ganikhodjaev and Razak, and also of Schonmann, and include GKS-type inequalities when the spin-space is taken as the set of $q$th roots of unity.
Planting and MCMC Sampling from the Potts model
本文通过种植技术分析Potts模型，证明在β_u以上随机聚类动力学混合快，并扩展了q,d的适用范围。
A Sharp Deconfinement Transition for Potts Lattice Gauge Theory in Codimension Two
本文将二维 Bernoulli 面片渗流的锐利相变结果推广到高维面片渗流及面片随机簇模型，从而在余维二的 Potts 格点规范理论中建立 Wilson 圈期望的锐利退禁闭相变，并发展了 i-面片随机簇模型的一般理论。
Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs
研究收敛到正则树的扩张图上Potts模型的临界极限，证明其局部弱极限为自由与连线Gibbs测度的混合，并推广了现有零场及偶数度数结果。
The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of…
The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q\geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where $\pc^\f (q) > \pu^\w (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculation…
The fuzzy Potts model in the plane: Scaling limits and arm exponents
该研究针对平面临界FK渗流染色得到的模糊Potts模型，证明其在共形不变标度极限假设下收敛于共形环系综变体，且离散与连续模型臂指数一致，进而确定了该模型的臂指数。
On the tractability of sampling from the Potts model at low temperatures via random-cluster dynamics
本文研究了一般图上低温Potts模型采样的最小结构性质，证明边渗流的强超临界相决定了random-cluster动力学的快速混合与指数级缓慢收敛。
Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory
本文建立 plaquette 随机簇模型与 Potts 晶格规范理论的 Wilson 圈精确对应，并证明环面模型在自对偶点存在同调逾渗尖锐相变。
We study graphical representations for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960's. The second model is the space-time percolation process, which is closely related to the contact model for the spread of disease. We consider a `space-time' random-cluster model and explore a range of useful probabilistic techniques for studying it. The space-time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase t…
Multi-point functions of a full-plane two-state fuzzy $4$-Potts model
通过研究全平面两态模糊 $4$-Potts 模型，证明了所有偶数多点自旋关联函数在适当归一化后收敛到显式的共形协变库仑气体型中性电荷和，并由此得到重整化磁化场的分布收敛及其矩。
Discontinuous transition in 2D Potts: I. Order-Disorder Interface convergence
本文研究 q > 4 时二维 Potts 模型在有序-无序边界条件下的界面，证明其具有 $\sqrt{N}$ 涨落并收敛于 Brownian 桥，方法基于 FK-percolation、six-vertex 模型与 Ashkin-Teller 模型随机团簇表示的耦合。
We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, for example, displacement, over such chains obey classical limit laws, and our construction leads to an effective random walk representation of percolation clusters. The results include a derivation of a sharp Ornstein--Zernike type asymptotic formula for two point functions, a proof of analyticity…
We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these pha…
The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a $Q$ -state Potts cluster, is solved in two dimensions. The dimension $\hat f(θ)$ of the boundary set with local wedge angle $θ$ is $\hat f(θ)=\fracπθ -\frac{25-c}{12} \frac{(π-θ)^2}{θ(2π-θ)}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\rm EP} =sup_θ\hat f(θ)$ of the external perimeter and $D_{\rm H}$ of the hull of a Potts cluster obey the duality equation $(D_{\rm EP}-1)(D_{\rm H}-1)={1/4}$. A related covariant MF …
We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, that updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the single-bond dynamics on the discrete $d$-dimensional torus of side length $L$ at the Potts transition temperature for $q$ large enough that are exponential in $L^{d-1}$, complementing a result of Borgs, Chayes and Tetali.
Critical exponents for planar random-cluster model with cluster-weight $q=4$
利用Baxter-Kelland-Wu耦合和六顶点模型高度函数收敛到高斯自由场，提取了$q=4$平面临界随机簇模型和四态Potts模型的临界指数。
In the Ising and Potts model, random cluster representations provide a geometric interpretation to spin correlations. We discuss similar constructions for the Villain and XY models, where spins take values in the circle, as well as extensions to models with different single site spin spaces. In the Villain case, we highlight natural interpretation in terms of the cable system extension of the model. We also list questions and open problems on the cluster representations obtained in this fashion.
For $Δ\ge 5$ and $q$ large as a function of $Δ$, we give a detailed picture of the phase transition of the random cluster model on random $Δ$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms …
We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.
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