Percolation, first-passage percolation, and covering times for Richardson's model on the n-cube

Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $Ω= n$ for $\mathbb{G} = \mathbb{Q}_n$ and $Ω= 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$ This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2})$ of Borgs, Chayes, van der …

Unoriented first-passage percolation on the n-cube

The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0, 1\}^n$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length $T_n$ of a path from $(0, 0, \dots, 0)$ to $(1, 1, \dots, 1)$ converges in probability to $\ln(1+\sqrt{2}) \approx 0.881$. It has previously been shown by Fill and Pemantle (1993) that this so-called first-passage time asymptotically almost surely satisfies $\ln(1+\sqrt{2}) - o(1) \leq T_n \leq 1+o(1)$, and …

广义汉明立方体上的自举渗流 第2卷

Bootstrap percolation on a generalized Hamming cube \MakeUppercase{\romannumeral 2}

本文扩展Balogh等人2009年的主定理，得到广义汉明立方体Q_{k,n}上阈值为N/2的自举渗流临界概率的一阶和二阶项。

Multi-scale Lipschitz percolation of increasing events for Poisson random walks

Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $τ$. Given a local event $E(i,τ)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $τ$, we prove the existence of a Lipschitz connected surface of cells $(i,τ)$ that separates the origin from infinity on w…

On the time constant of high dimensional first passage percolation

We study the time constant $μ(e_{1})$ in first passage percolation on $\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $$\lim_{d \to \infty} \frac{μ(e_{1}) d}{\log d} = \frac{1}{2a},$$ where $a \in [0,\infty]$ is a constant that depends only on the behavior of the distribution of the passage times at $0$. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a $d$-dimensional cube or diamond, provided that $d$ is large enough.

Differentiability at the edge of the percolation cone and related results in first-passage percolation

We study first-passage percolation in two dimensions, using measures mu on passage times with b:=inf supp(mu) >0 and mu({b})=p \geq p_c, the threshold for oriented percolation. We first show that for each such mu, the boundary of the limit shape for mu is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if mu is not purely atomic the graph of infection has infinitely many ends. We go on to…

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.

最后通行渗流中最可能的测地线

On the most likely geodesic in last passage percolation

本研究探讨最后通行渗流中最可能的测地线问题，确定三种利于极端角路径的机制，在指数最后通行渗流中证明角路径近乎典型，并推测测地线概率单调性在一般情况下成立。

The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n…

敌对环境中的首次通过渗流不具有单调性

First passage percolation in hostile environment is not monotone

该研究针对双竞争首次通过渗流过程的存活概率单调性问题，构造特殊图证明其存活概率并非参数μ或λ的单调函数，存在多相变，与2型Richardson模型等生长过程行为不同。

Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process

We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at…

Cluster size in bond percolation on the Platonic solids

The main objective of this paper is to study the size of a typical cluster of bond percolation on each of the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Looking at the clusters from a dynamical point of view, i.e., comparing the clusters with birth processes, we first prove that the first and second moments of the cluster size are bounded by their counterparts in a certain branching process, which results in explicit upper bounds that are accurate when the density of open edges is small. Using that vertices surrounded by closed edges …

通过耦合技术得到指数型最后通过渗流中退出点的最优阶界

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

该研究提出一种新概率方法，在指数权重最后通过渗流模型中推导出测地线退出点的最优三次指数阶界，拓展了耦合方法的应用范围，还得到了测地线涨落等问题的相关上界结果。

末行渗流中从稳定性到混沌的转变

From stability to chaos in last-passage percolation

本文研究动态末行渗流模型中稳定性到混沌的相变，证明在时间参数与通行时间方差除以规模相关的临界点处，通行时间相关性发生突变且测地线由高度重合变为几乎不相交。

促进排斥过程

Facilitated exclusion process

本研究针对一维Facilitated TASEP粒子系统，通过半象限最后通过渗流映射与Pfaffian Schur过程求解，证明其最右粒子位置服从GSE统计，揭示了带密度间断的排斥过程的立方根涨落特征及模型依赖的统计分布，并探讨了三类分布的二维交叉现象。