Yacine Aoun
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Fluctuation theory of connectivities for subcritical random cluster models
我们发展了一个亚临界随机团簇模型的连通性涨落理论。该理论是基于一个全面的非微扰概率描述的长连接集群基本上是一维链的不可约对象。局部观测量的统计，例如，位移，在这样的链遵守经典的极限定律，我们的建设导致一个有效的随机游动表示的渗流集群。结果包括两点函数的一个尖锐的Ornstein-Zernike型渐近公式的推导，反相关长度的解析性和严格凸性的证明，以及扩散标度下连通团簇的一个不变原理的证明.在两个维度的对偶性的考虑，使这些结果的超临界最近邻随机集群措施，特别是最近邻Potts模型的相变制度。相应地，我们证明了在二维Potts平衡晶体形状总是解析的和严格凸的，并且不同相之间的界面总是扩散的。因此，没有粗糙化过渡是可能的，在整个制度，我们的结果适用。我们的结果在有限体积有线连接指数衰减的假设下（假设（1.2）），在矩形域中，这是假设在整个亚临界状态下，后者是真实的，在任何维度，当$q=1$，$q=2$，当$q$是足够大的。在二维中，假设（1.2）在无限体积测度中的连通性指数衰减时成立。通过对偶性，这包括所有有序相之间具有正表面张力的超临界最近邻Potts模型。
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…
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…
Sharp mixing time asymptotics of Glauber dynamics for the Curie-Weiss-Potts model at low temperatures
本文研究了Curie-Weiss-Potts模型在低温下的Glauber动力学混合时间渐进行为，揭示了元稳态理论在分析缓慢混合中的核心作用，并证明了无截止现象。
Subcritical sharpness for real-valued spin models
本文针对一般传递图上的实值自旋模型，利用随机簇表示推广了OSSS不等式，证明了次临界区域中关联函数指数衰减。
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。
We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernandez, den Hollander and Martınez for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and u…
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…
Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs
研究了一般拟传递图上的齐次独立渗流问题。我们证明，在无序制度中，所有的集群是有限的几乎肯定，事实上，集群大小的期望是有限的。这将Menshikov和Aizenman & Barsky的一个著名定理扩展到所有准传递图。此外，我们推断，在这种无序制度的集群大小分布指数衰减，扩展了Aizenman和Newman的结果。我们的研究结果适用于边缘和网站的渗流，以及远程（边缘）渗流。证明是基于修改的Aizenman & Barsky方法。
We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $μ$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $μ$ with a countable number of scaled independent copies. The $n$-th copy is a Boolean percolation with radius distribution $μ|_{[1,κ]}$ rescaled by $κ^{n}$. We prove that under some regularity assumption on $μ$, the subcritical phase of the multiscale model is sharp for $κ$ large enough. Moreover, we prove that the existence of an unbounded connected component depends only on t…
Sharp freezing time estimates for the subcritical Facilitated Exclusion Process
研究一维环上促进排斥过程（FEP）在次临界密度下的瞬态时间（冻结时间），证明其尺度为Θ(log³ N)，并在近临界情况下呈多项式尺度N^{1∧(2α)}。
The left tail of the subcritical derivative martingale in a branching random walk
本文在高斯分支随机游走框架下，精确估计了$L^4$相中导数鞅分布的左尾，回答了Lacoin等人提出的猜想。
Renormalization of crossing probabilities in the dilute Potts model
该研究通过四重划分方法分析稀释Potts模型的交叉概率行为，揭示其在不同临界条件下可能的四种行为范围。
Rapid phase ordering for Ising and Potts dynamics on random regular graphs
该研究针对随机d-正则图上低温Ising与q态Potts Glauber动力学混合慢的问题，证明带初始偏差时可在O(log n)时间内准平衡到条件平稳分布，并通过引入耦合非马尔可夫刚性动力学解决了低温下负信息传播的控制难题。
Sharp phase transition and critical behaviour in 2D divide and colour models
考虑具有固定参数p<p_c的亚临界伯努利键渗流。我们定义了一个依赖的网站渗透模型，通过以下程序：对于每个键簇，我们的颜色在集群中的所有顶点黑色的概率r和白色的概率1-r，彼此独立。在正方形格点上，分别定义了位模型及其对偶的临界概率r_c（p）和r_c^*（p），我们证明了对于所有亚临界p，r_c（p）+r_c^*（p）=1。在三角形格点上，我们的方法也适用，这导致了对于所有亚临界p，r_c（p）=1/2。在两个格点上，我们得到了在r_c（p）以下团簇尺寸的指数衰减，在r_c（p）处平均团簇尺寸的发散，以及在[0，1]上的渗流函数在r中的连续性.我们还讨论了我们的结果的可能的扩展，并制定了一些自然的代数。我们的方法依赖于对偶性的考虑和最近扩展的经典RSW定理。
We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). More precisely, we prove that: -In the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius dist…
We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington-Kirkpatrick model with vector spins interacting throug…
Uniform analyticity of local observables in FK-percolation and analyticity of the Ising spontaneous magnetisation
论文证明了在FK渗流模型中，局部事件的概率在渗流参数p下均匀解析，并满足指数增长界，从而证明了Potts模型磁化率在合适参数范围内的解析性，包括伊辛模型在三维及以上所有超临界区域的解析性。
We present a simple criterion, only based on second moment assumptions, for the convergence of polynomial or Wiener chaos to a Gaussian limit. We exploit this criterion to obtain new Gaussian asymptotics for the partition functions of two-dimensional directed polymers in the sub-critical regime, including a singular product between the partition function and the disorder. These results can also be applied to the KPZ and Stochastic Heat Equation. As a tool of independent interest, we derive an explicit chaos expansion which sharply approximates the logarithm of the partition function.
The purpose of this modest note is to provide a short proof of the sharpness of the phase transition for the Random-cluster model with $q=2$ by extending the approach developed by Duminil-Copin and Tassion for $q=1$. This in particular implies the exponential decay of the two point-correlation function in the subcritical Ising model.
The Random-Cluster Model
随机簇模型是对概率和统计物理有重要意义的各种随机过程的统一，包括渗流模型、伊辛模型和波茨模型;此外，它们的研究对某些随机组合结构和电网络的理论产生了影响。伊辛/波茨模型的大部分（但不是全部）物理理论最好在随机簇表示的上下文中实现。这一系统的总结随机集群模型包括帐户的基本方法和不等式，唯一性和规格的无限体积措施，存在和性质的相变，和结构的亚临界和超临界相。二维晶格的理论比三维和更多维的理论发展得更好。有一个开放的问题，包括一些重大意义的无序系统的一般领域的丰富集合，这些都是突出时遇到。在主要的未决问题中，有一个问题是确定的确切性质的相变的一般值的集群加权因子q，和问题的证明，临界随机集群模型在二维，1 \le q \le 4，收敛时重新缩放到一个随机Löwner演化。总的来说，重点是随机集群模型本身，而不是在其应用程序的伊辛和波茨系统。
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。
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensiona…
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $π_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $π_p(n) / π_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" fr…
The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\ge 4$, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respect…
Left Tail of the Derivative Martingale in a Gaussian BRW in the Entire Subcritical Regime
研究高斯分支随机游走在亚临界区域中导数鞅极限的左尾概率，通过建立双边估计，证实了特定猜想。
A random field $X = (X_v)_{v \in G}$ on a quasi-transitive graph $G$ is a factor of i.i.d. if it can be written as $X=φ(Y)$ for some i.i.d. process $Y= (Y_v)_{v \in G}$ and equivariant map $φ$. Such a map, also called a coding, is finitary if, for every vertex $v \in G$, there exists a finite (but random) set $U \subset G$ such that $X_v$ is determined by $\{Y_u\}_{u \in U}$. We construct a coding for the random-cluster model on $G$, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström--Jonasson--Lyons. We also prove that th…
On the two-point function of the Potts model in the saturation regime
本文研究无限程 Random-Cluster/Potts 模型，给出非平凡饱和区域存在的最优判据，并证明该区域内两点函数的非 Ornstein–Zernike 尖锐渐近及相关相变性质。
Occupation times of subcritical branching immigration system with Markov motions
我们考虑一个分支系统的粒子运动根据一个马尔可夫家庭在$\Rd$和经历亚临界分支与恒定速率$V>0$。新粒子按照齐性空间-时间泊松随机场随机场向系统迁移。在马尔可夫族的假设下，研究了重标占有时间的涨落过程。在此一般性条件下，证明了一个泛函中心极限定理.分支律的亚临界性对于极限行为是至关重要的，并且在某种意义上描述了粒子运动的性质。正是由于这个原因，极限对于所有维度都是相同的，并且可以对广泛的一类马尔可夫过程获得。另一个结果是极限$\SP$值Wiener过程的形式，具有简单的时间结构和复杂的空间结构。这种行为与临界分支系统的情况形成鲜明对比。
We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.
Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights
研究稀疏秩1随机图上q态Potts模型中决定临界逆温度βc的关键量tc，通过构建定义方程分析tc、tc'和tc''，得出相关上界及大q行为对应情况，证明这些界在特定极限下是紧的。
Sharp Asymptotics for the Largest Component in the Subcritical Regime of Preferential Attachment Without Vertex Growth
研究Pittel无顶点增长优先连接过程中最大分量大小，在亚临界区域给出精确渐近公式，常数随参数变化并趋于Erdős–Rényi值。
We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $β<β_c$, and the mean-field lower bound $\mathbb{P}_β[0\longleftrightarrow\infty]\ge (β-β_c)/β$ for $β>β_c$. For finite-range models, we also prove that for any $β<β_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we pro…
We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…
Existence of subcritical percolation phases for generalised weight-dependent random connection models
本文研究带标记泊松点过程上的广义权重依赖随机连接模型，通过控制长边给出临界环带穿越强度为正及次临界相存在的充分条件，并建立典型簇直径与规模的尾界。
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。
Cluster sizes in subcritical soft Boolean models
研究软布尔模型中亚临界渗流相典型最大连通分量的欧几里得直径和点数尾行为，基于边权和半径尾指数给出精确判据。
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。
Asymptotics of the $ϕ^4_1$ measure in the sharp interface limit
本文研究 $ϕ^4_1$ 测度在尖锐界面极限下的渐近行为，在亚临界区域证明了大偏差原理并给出跃迁次数的精确渐近，在超临界区域证明界面渐近服从泊松点过程。
Graphical representations of Ising and Potts models
我们研究两个相关模型的图形表示。第一个模型是横向场量子伊辛模型，是由Lieb，Schultz和Mattis在20世纪60年代引入的原始伊辛模型的扩展。第二个模型是时空渗流过程，它与疾病传播的接触模型密切相关。我们考虑一个“时空”随机簇模型，并探讨了一系列有用的概率技术研究it. The时空Potts模型出现作为一个自然的推广的量子伊辛模型。在这些模型中的相变的基本性质的治疗，如事实上，有至多一个无界FK-集群，并由此产生的下界上的临界值$\ZZ$。我们还开发了另一种图形表示的量子伊辛模型，称为随机宇称表示。这种表示是基于经典伊辛模型的随机电流表示，并允许我们更详细地研究相变和临界行为。随机宇称表示被用来证明量子伊辛模型中相变的尖锐性，并建立一些临界指数的界限。我们解决这些问题，通过使用随机奇偶表示建立某些微分不等式，积分的结果。最后，我们探讨了上述结果的一些后果和可能的扩展。例如，我们确定的临界点的量子伊辛模型在$\ZZ$和“星状”几何。
In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$ with polynomial size window. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality, inspired by the recent developments by Ahlberg, Broman, Griffiths, and Morris and by …
The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $Λ_{n}$ of the integer lattice graph $\mathbb Z^2$, where the model exhibits a sharp phase transition at $p=p_c(q)$. Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of $Λ_n$, known as boundary conditions. For select boundary conditions that …
Markovian renormalisation for percolation in high-dimension: Semi-decidability of mean field behavior
针对高维（d>6）伯努利逾渗，提出用马尔可夫链近似开放路径以传递两点函数信息，证明临界平均场行为半可判定，并给出磁化率、回转半径渐近及局部中心极限定理。
Sharpness of the avalanche phase transition in the Bak--Sneppen model
该研究证明了一维Bak-Sneppen模型中两种更新规则下雪崩相变的尖锐性，确定了雪崩临界阈值并验证了猜想的平稳乘积极限。
Subcritical annulus crossing in spatial random graphs
研究空间随机图中临界亚临界环交叉强度，通过环交叉与长边出现的关系推导条件，探讨模型中环交叉的机制及长边概率衰减的影响。
Sharpness of the percolation phase transition for weighted random connection models
本文研究加权随机连接模型中渗透相变的锐度，通过证明亚临界和超临界区域的簇大小分布特性，揭示了临界行为的数学结构。
On strong sharp phase transition in the random connection model
本文研究了随机连接模型中强尖锐相变现象，推导了临界强度下的聚类性质和渗透均场界，扩展了相关研究成果。
Typical sofic entropy and local limits for free group shift systems
本文研究自由群移位系统不变测度的典型上下 sofic 熵，给出其等于退火熵的条件，并用于随机正则图 Gibbs 态局部极限及 Ising 模型熵相变。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。