Federico Camia, Daniel L. Stein
In the introduction to this volume, we discuss some of the highlights of the research career of Chuck Newman. This introduction is divided into two main sections, the first covering Chuck's work in statistical mechanics and the second his work in percolation theory, continuum scaling limits, and related topics.
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The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $ρ/n$ for some $ρ>0$ constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as $\frac1λ\log n$ for a $λ>0$ and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the sho…
We pursue the study of a random coloring first passage percolation model introduced by Fontes and Newman. We prove that the asymptotic shape of this first passage percolation model continuously depends on the law of the coloring. The proof uses several couplings, particularly with greedy lattice animals.
We study first-passage percolation on Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 90's by C. Newman and collaborators under strong assumptions on the limiting shape and weight distribution. In this paper we develop a framework for working with distributional limits of Busemann functions and use it to prove forms of Newman's results under minimal assumptions. For instance, we show a form of coalescence of long finite geodesics in any determinist…
In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity $τ_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.
A discontinuous percolation phase transition on the hierarchical lattice
该研究在分层晶格上证明了不连续相变的阈值，并提出分层版本的$M^2β=1$猜想。
We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesi…
This is a short survey of work on percolation and first-passage percolation since the publication (in 1996 and 1984, respectively) of the two authors' Saint-Flour notes on these topics.
We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic defined on a configuration model random graph, in which infective individuals can infect only their neighbours in the graph during an infectious period which has an arbitrary but specified distribution. Central limit theorems for the final size (number of initial susceptibles that become infected) of such an epidemic as the population size $n$ tends to infinity, with explicit, easy to compute expressions for the asymptotic variance, are proved assuming that the degrees are bounded. The results are obtained for …
Second look at the spread of epidemics on networks
本文指出 Newman 提出的 SIR 模型与键渗流模型的同构是不正确的，提出了在任何有限群体中与 SIR 模型完全同构的流行病渗流网络，并证明在存在非退化感染期分布时键渗流模型预测失效。
In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semi-infinite geodesics in some fixed direction $α$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by Wüthrich. These results will be crucial in the development of an upcoming paper on the relatio…
Coexistence of infinite clusters for percolation and Ising model on $\mathbb{Z}^d$
本文证明在 $\mathbb{Z}^d$ 上，当维度足够高且参数接近临界值时，渗流和 Ising 模型的无限簇补集或 $+$ 自旋随机占优超临界位点渗流，改进了已有结果。
Percolation on Finite Graphs
本讲义基于特拉维夫大学研究生课程，系统介绍有限图上的渗流理论，涵盖随机图相变、巨分量、长路径、连通性与完美匹配阈值及随机超立方体中的相关结果。
Critical points and singularities are encountered in the study of critical phenomena in probability and physics. We present recent results concerning the values of such critical points and the nature of the singularities for two prominent probabilistic models, namely percolation and the more general random-cluster model. The main topic is the statement and proof of the criticality and universality of the canonical measure of bond percolation on isoradial graphs (due to the author and Ioan Manolescu). The key technique used in this work is the star--triangle transformation, known also as the Ya…
On the influence of edges in first-passage percolation on $\mathbb{Z}^d$
本文研究 $\mathbb{Z}^d$ 上首达渗流中位于测地线上的边的数量，证明其一致有界，并给出定量上界，解决了 Benjamin-Kalai-Schramm 的问题，同时加强了横向波动下界。
There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$ and adapt the results of \cite{boivin1990first} to prove a shape theorem for ergodic random pseudometrics on $\mathbb R^d$. A natural application of this result will be the study of FPP on random tessellations where a fluid starts in the zero cell and takes a random time to pass through the boundary of a cell into a neighbouring cell. We find that a tame rand…
We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics. A complete bibliography is included of his publications.
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