arXiv subjects
Publications and source records attributed to Harry Kesten.
We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.
Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.
The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on $\Z^d$, $d \geq 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density $μ$. The green particle also jumps at rate 1, but uses different transition kernels $p'$ and $p''$ depending on whether it sees a red particle or not. It is shown that, in the limit as $μ\to\infty$, the speed of the green particle tends to the average jump under $p'$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in \cite{KeSi} to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space-time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.
On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+ \colon y=i}$ are first declared to be bad or good with probabilities $\de$ and $1-\de$ respectively, independently of each other. Given the configuration of lines, sites on good lines are open with probability $p_{_G}>p_c$, the critical probability for the standard oriented site percolation on $\mathbb Z_+ \times \mathbb Z_+$, and sites on bad lines are open with probability $p_{_B}$, some small positive number, independently of each other. We show that given any pair $p_{_G}>p_c$ and $p_{_B}>0$, there exists a $δ(p_{_G}, p_{_B})>0$ small enough, so that for $δ\le δ(p_G,p_B)$ there is a strictly positive probability of oriented percolation to infinity from the origin.
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.
An ordered pair of semi-infinite binary sequences $(η,ξ)$ is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from $η$ and zeroes from $ξ$, whichwould map both sequences to the same semi-infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: $η$ and $ξ$ being independent i.i.d. Bernoulli sequences with parameters $p^\prime$ and $p$ respectively, does it exist $(p', p)$ so that the set of compatible pairs has positive measure? It is known that this does not happen for $p$ and $p^\prime$ very close to 1/2. In the positive direction, we construct, for any $ε> 0$, a deterministic binary sequence $η_ε$ whose set of zeroes has Hausdorff dimension larger than $1-ε$, and such that $\mathbb{P}_p {ξ\colon (η_ε,ξ) \text {is compatible}} > 0$ for $p$ small enough, where $\mathbb{P}_p$ stands for the product Bernoulli measure with parameter $p$.
We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time $t$, we have an "aggregate" consisting of $\Bbb{Z}\cap[0,R(t)]$ [with $R(t)$ a positive integer]. We also have $N(i,t)$ particles at $i$, $i>R(t)$. All these particles perform independent continuous-time symmetric simple random walks until the first time $t'>t$ at which some particle tries to jump from $R(t)+1$ to $R(t)$. The aggregate is then increased to the integers in $[0,R(t')]=[0,R(t)+1]$ [so that $R(t')=R(t)+1$] and all particles which were at $R(t)+1$ at time $t'{-}$ are removed from the system. The problem is to determine how fast $R(t)$ grows as a function of $t$ if we start at time 0 with $R(0)=0$ and the $N(i,0)$ i.i.d. Poisson variables with mean $μ>0$. It is shown that if $μ<1$, then $R(t)$ is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured that $R(t)$ will grow linearly in $t$ if $μ$ is large enough.
Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $π\subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with the last coordinate increasing along the path, and of length $n$. Define for such paths $W(π) = \text{number of vertices $π_i, 1 \le i \le n$, with}X(π_i) = e^b$. Finally let $N_n(\al) = \text{number of paths $π$ of length $n$ starting at $π_0 = \bold 0$ and with $W(π) \ge \al n$.}$ We establish several properties of $\lim_{n \to \infty} [N_n]^{1/n}$.
Let G be a vertex transitive graph. A study of the range of simple random walk on G and of its bridge is proposed. While it is expected that on a graph of polynomial growth the sizes of the range of the unrestricted random walk and of its bridge are the same in first order, this is not the case on some larger graphs such as regular trees. Of particular interest is the case when G is the Cayley graph of a group. In this case we even study the range of a general symmetric (not necessarily simple) random walk on G. We hope that the few examples for which we calculate the first order behavior of the range here will help to discover some relation between the group structure and the behavior of the range. Further problems regarding bridges are presented.
We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on $\mathbb{Z}^d$, with jump rate $D_A$. These particles are called $A$-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-),x\in\mathbb{Z}^d$, are i.i.d., mean-$μ_A$ Poisson random variables. In addition, there are $B$-particles which perform continuous-time simple random walks with jump rate $D_B$. We start with a finite number of $B$-particles in the system at time 0. $B$-particles are interpreted as individuals who have heard a certain rumor or who are infected. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if $\widetilde{B}(t):=\{x\in\mathbb{Z}^d:$ a $B$-particle visits $x$ during $[0,t]\}$ and $B(t)=\widetilde{B}(t)+[-1/2,1/2]^d$, then we investigate the asymptotic behavior of $B(t)$. Our principal result states that if $D_A=D_B$ (so that the $A$- and $B$-particles perform the same random walk), then there exist constants $0<C_i<\infty$ such that almost surely $\mathcal{C}(C_2t)\subset B(t)\subset \mathcal{C}(C_1t)$ for all large $t$, where $\mathcal{C}(r)=[-r,r]^d$. In a further paper we shall use the results presented here to prove a full ``shape theorem,'' saying that $t^{-1}B(t)$ converges almost surely to a nonrandom set $B_0$, with the origin as an interior point, so that the true growth rate for $B(t)$ is linear in $t$. If $D_A\ne D_B$, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_1t)$ eventually.
We show that a certain model for the spread of an infection has a phase transition in the recuperation rate. The model is as follows: There are particles or individuals of type A and type B, interpreted as healthy and infected, respectively. All particles perform independent, continuous time, simple random walks on Z^d with the same jump rate D. The only interaction between the particles is that at the moment when a B-particle jumps to a site which contains an A-particle, or vice versa, the A-particle turns into a B-particle. All B-particles recuperate (that is, turn back into A-particles) independently of each other at a rate lamda. We assume that we start the system with N_A(x,0-) A-particles at x, and that the N_A(x,0-), x in Z^d, are i.i.d., mean mu_A Poisson random variables. In addition we start with one additional B-particle at the origin. We show that there is a critical recuperation rate lambda_c > 0 such that the B-particles survive (globally) with positive probability if lambda < lamda_c and die out with probability 1 if lambda > \lamda_c.
We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous time simple random walk on the d-dimensional lattice. These particles are called A-particles and move independently of each other. We assume that we start the system with a Poisson number of particles at each lattice site x, with the number of particles at different x's i.i.d. In addition, there are a finite number of B-particles which perform the same continuous time simple random walks as the A-particles. A- and B-particles are interpreted as individuals who are healthy or infected, respectively. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. Let B(t) be the set of sites visited by a B-particle during [0,t]. We show that B(t) grows linearly in time and has an asymptotic shape; more precisely, there exists a non-random convex, compact set B_0 such that almost surely, for all 0 < a <1, (1-a)tB_0 is contained in B(t) and B(t) is contained in (1+a)tB_0 eventually.
The uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle proved that the USF consists a.s. of a single tree if and only if d <= 4. We prove that any two components of the USF in Z^d are adjacent a.s. if 5 <= d <= 8, but not if d >= 9. More generally, let N(x,y) be the minimum number of edges outside the USF in a path joining x and y in Z^d. Then a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.
We describe the percolation model and some of the principal results and open problems in percolation theory. We also discuss briefly the spectacular recent progress by Lawler, Schramm, Smirnov and Werner towards understanding the phase transition of percolation (on the triangular lattice).
This paper contains the proofs of Theorems 2 and 3 of the article entitled Random Electrical Networks on Complete Graphs, written by the same authors and published in the Journal of the London Mathematical Society, vol. 30 (1984), pp. 171-192. The current paper was written in 1983 but was not published in a journal, although its existence was announced in the LMS paper. This TeX version was created on 9 July 2001. It incorporates minor improvements to formatting and punctuation, but no change has been made to the mathematics. We study the effective electrical resistance of the complete graph $K_{n+2}$ when each edge is allocated a random resistance. These resistances are assumed independent with distribution $P(R=\infty)=1-n^{-1}γ(n)$, $P(R\le x) = n^{-1}γ(n)F(x)$ for $0\le x < \infty$, where $F$ is a fixed distribution function and $γ(n)\toγ\ge 0$ as $n\to\infty$. The asymptotic effective resistance between two chosen vertices is identified in the two cases $γ\le 1$ and $γ>1$, and the case $γ=\infty$ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.