Surface order large deviations for 2d FK-percolation and Potts models

Large deviations for directed percolation on a thin rectangle

Following the recent investigations of Baik and Suidan in \cite{baik2005gcl} and Bodineau and Martin in \cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\mathbb{Z}^{2}_{+}$ whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in \cite{baik2005gcl} and \cite{bodineau2005upl}, on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.

Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation.

Moderate deviation and exit time estimates for stationary Last Passage Percolation

We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on …

FK渗流中局部可观测量的均匀解析性及伊辛自发磁化率的解析性

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。

Upper tail large deviations for a class of distributions in First-passage percolation

In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.

Moderate deviations for the chemical distance in Bernoulli percolation

In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.

Right-tail moderate deviations in the exponential last-passage percolation

We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the increment-stationary process. The arguments utilize calculations with the stationary version and a moment generating function identity due to E. Rains, for which we give a short probabilistic proof. As applications of the deviation bounds, we derive upper bounds on the speed of distributional convergence in the Busemann function and competition interface limits.

Large deviations for the chemical distance in supercritical Bernoulli percolation

The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

余维二中 Potts 格点规范理论的锐利退禁闭相变

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。

Large deviation bounds for the volume of the largest cluster in 2D critical percolation

Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C x^(2/alpha)) for x > 1 and large n with alpha = 5/48 and C > 0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d > 2.

单点上大偏差事件下的条件指数有向最后通过渗流

Conditional exponential directed last passage percolation under a one-point upper large deviation event

探讨有向最后通过渗流模型在单点上大偏差事件下最后通过时间场的变化，通过分析显式多点分布，证明条件大数定律并计算极限涨落。

Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $Ω$ and escape from $Ω$. A cutset is a set of edges that separates $Γ^1$ from $Γ^2$ in $Ω$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $Ω$ and the distribution of the capacities of the edges, we already know a law of large numbers fo…

Spins, percolation and height functions

To highlight certain similarities in combinatorial representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters. To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of diffe…

Lower large deviations for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behaviour of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviat…

Large deviation principle for the streams and the maximal flow in first passage percolation

We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R ^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the source and the sink, i.e., where the water can enter in $Ω$ and escape from $Ω$. A maximal stream is a vector measure $\overrightarrowμ_n^{max}$ that describes how the maximal amount of fluid can enter through $Γ^1$ and spreads in $Ω$. Under some assumptions on $Ω$ and $G$, we already know a law of large number for $\overrightarrowμ_n^{max}$. The s…

Upper large deviations for the maximal flow in first passage percolation

We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $ϕ_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $ϕ_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $ν$. We look now at the probability that $ϕ_{n^{d-1},h(n)} / n^{d-1}$ is greater than $ν+ ε$ for some $ε>0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder…

首达渗流中随机度量在速度 $n$ 下的大偏差原理

Large deviation principle at speed $n$ for the random metric in first-passage percolation

本文研究 $\mathbb Z^d$ 标准首达渗流中盒内重标度随机度量的下尾大偏差，证明其在指数矩条件下满足速度 $n$ 的大偏差原理，并给出速率函数的三种表达及弱矩条件下的概率估计。

Quenched large deviations for simple random walks on percolation models including long-range correlations

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffus…

平面上的模糊Potts模型：标度极限与臂指数

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。

A simple renormalization flow for FK-percolation models

We present a setup that enables to define in a concrete way a renormalization flow for the FK-percolation models from statistical physics (that are closely related to Ising and Potts models). In this setting that is applicable in any dimension of space, one can interpret perturbations of the critical (conjectural) scaling limits in terms of stationary distributions for rather simple Markov processes on spaces of abstract discrete weighted graphs.

关于超临界渗流中化学距离上尾大偏差速率函数的研究

On the upper tail large deviation rate function for chemical distance in supercritical percolation

本文研究了超临界键渗流中化学距离的上尾大偏差问题，证明了当 $d\ge 3$ 且 $\varepsilon>0$ 足够小时速率函数的存在性，并揭示该大偏差事件由时空切点迫使测地线绕行引发。

Lower Deviations in $β$-ensembles and Law of Iterated Logarithm in Last Passage Percolation

For the last passage percolation (LPP) on $\mathbb{Z}^2$ with exponential passage times, let $T_{n}$ denote the passage time from $(1,1)$ to $(n,n)$. We investigate the law of iterated logarithm of the sequence $\{T_{n}\}_{n\geq 1}$; we show that $\liminf_{n\to \infty} \frac{T_{n}-4n}{n^{1/3}(\log \log n)^{1/3}}$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (J. Theor. Probab., 2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight …

一维长程模型再探讨

Long-range models in 1D revisited

本文重新审视了一维长程 percolation、Ising model 和 Potts models 的经典结果，证明了在大 $β$ 下 $1/r^2$-相互作用存在对称性破缺且相变必然不连续，并得出对所有 $q\geq 1$ 均有 $β^*(q)=1$。

Upper large deviations for the maximal flow through a domain of $\bolds{\mathbb{R}^d}$ in first passage percolation

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviatio…

末次穿越渗流模型的大偏差原理

Large Deviation Principle for Last Passage Percolation Models

本文应用并改进已有框架，在不依赖精确可解性的条件下为末次穿越渗流（LPP）模型建立了度量层面的大偏差原理，并据此推导出测地线的大偏差原理，为研究测地线大横向涨落提供了简化方法。

Large-deviations/thermodynamic approach to percolation on the complete graph

We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are ``small.'' One consequence is an immediate derivation of the ``cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no …

Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $τ$ between the lower half and the upper half of the box, and the maximal flow $ϕ$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $τ$ in rational directions. Kesten and Zhang have proved the law of large numbers for $τ$ and $ϕ$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same de…

A large deviation approach to super-critical bootstrap percolation on the random graph $G_{n,p}$

We consider the Erdös--Rényi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible rang…

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{κ'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on d…

二维渗流中等周常数的大偏差

Large deviations for the isoperimetric constant in 2D percolation

本文研究二维超临界渗流巨分支等周常数的精细渐近行为，确定其大偏差概率，并揭示下尾区域中表面阶与体积阶之间的相变。

首达渗流中随机度量在速度$n^d$下的大偏差原理

Large deviation principle at speed $n^d$ for the random metric in first-passage percolation

该研究针对$\u2124^d$上的标准首达渗流模型，通过多维次可加性论证等方法，证明了紧集上重整化首达时间满足速度为$n^d$的大偏差原理，并推导了相关推论。

二维 Potts 模型中的不连续相变：I. 有序-无序界面的收敛性

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。

Explicit isoperimetric constants and phase transitions in the random-cluster model

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…

超临界基于核的空间随机图中巨分量的大偏差

Large deviations of the giant in supercritical kernel-based spatial random graphs

本文研究超临界空间随机图中巨分量大小的大偏差，证明长程边可将下尾多项式速度指数从(d-1)/d提高到任意ζ_*∈((d-1)/d,1)，并刻画了上尾在幂律度分布下的对数速度及度齐次模型中的线性速度。

随机图上亚临界bootstrap渗流的大偏差

Large deviations for subcritical bootstrap percolation on the random graph

针对Erdős-Rényi随机图上的亚临界bootstrap渗流，本文计算了小规模初始感染集感染异常多顶点事件的速率函数，确定了对应大偏差的最小代价轨迹。

指数末通过渗流中测地线中点与端点的尖锐偏差界

Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation

本文研究指数末通过渗流中测地线在端点与中点处非典型横向涨落的概率，得到两个尖锐偏差界，并解决了Liu提出的一个猜想的特殊情形。

Upper Tail Large Deviations in First Passage Percolation

For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as $\exp (-Θ(n^2))$. However the question of existence of the rate function i.e., whether the log-probability normalized by $n^2$ tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the…

单点上大偏差事件条件下的指数定向最后通过渗流的条件平坦指数模型

Conditional Flat Exponential Directed Last Passage Percolation under a One-Point Upper Large Deviation Event

该论文研究指数定向最后通过渗流在单点上大偏差事件下的条件模型，提出条件极限形状的预测并证明波动结果，发现对角线附近波动由共享高斯分量的布朗桥描述。

Quenched Large Deviations for Simple Random Walks on Supercritical Percolation Clusters

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster on $\Z^d$, $d\geq 2$.. We take the point of view of the moving particle and first prove a quenched LDP for the distribution of the {\it{pair empirical measures}} of the environment Markov chain. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit (variational) formulas. Our results are based on invoking ergodicity arguments in this non-elliptic set up to contr…

A variational formula for large deviations in First-passage percolation under tail estimates

Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally d…

几何最后通过渗流中的精确大偏差

Precise large deviations in geometric last passage percolation

本文研究几何 LPP 的最后通过时间，通过建立与 JUE 和 TUE 最大特征值相关的两个对偶公式，精确刻画上下尾大偏差概率至常数项，并得到 TUE 特征多项式矩的渐近性。

Small deviation estimates and small ball probabilities for geodesics in last passage percolation

For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $Γ_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $Γ_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $Γ_{n}$ and establish that for small $δ$, the probability that $Γ_{n}$ is contained in a strip of width $δn^{2/3}$ around the diagonal is $\exp (-Θ(δ^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the …

具有一般 i.i.d. 权重的最后到达渗流中测地线中点波动的大偏差

Large deviations of geodesic midpoint fluctuations in last-passage percolation with general i.i.d. weights

本文在一般最后到达渗流模型中建立了测地线中点横向波动的大偏差极限，并用最后到达值的右尾大偏差速率函数和形状函数表示了速率函数，验证了指数权重下的相关猜想。

Linear speed large deviations for percolation clusters

Let C_n be the origin-containing cluster in subcritical percolation on the lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact, connected, origin-containing subsets of R^d, endowed with the Hausdorff metric delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to \infty}(1/n) \log P(C_n \in Γ) = -\inf_{S \in Γ} λ(S) where lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given p…

有界权重下第一通过渗流的中等偏差

Moderate deviations in first-passage percolation for bounded weights

本文研究了有界权重下第一通过渗流的中等和大偏差，推导了相关概率的指数衰减形式，并揭示了波动与大偏差衰减率之间的关系。