Coalescing and annihilating random walk with ‘action at a distance’

We consider a system of particles which perform continuous time random walks onZ ^d . These random walks are independent as long as no two particles are at the same site or adjacent to each other. When a particle jumps from a site x to a sitey and there is already another particle aty or at some neighbory′ ofy, then there is an interaction. In the coalescing model, either the particle which just jumped toy is removed (or, equivalently, coalesces with a particle aty ory′) or all the particles at the sites adjacent toy (other thanx) are removed. In the annihilating random walk, the particle which just jumped toy and one particle aty ory′ annihilate each other. We prove that when the dimensiond is at least 9, then the density of this system is asymptotically equivalent toC/t for some constant C, whose value is explicitly given.     

Slowdown estimates and central limit theorem for random walks in random environment

This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ℤ ^d , when d≥2. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in the investigation of both problems. This article also improves the previous work of the author [24], concerning estimates of probabilities of slowdowns for walks which are neutral or biased to the right. Received May 31, 1999 / final version received January 18, 2000?Published online April 19, 2000     

Large deviations for the symmetric simple exclusion process in dimensions d≥ 3

We consider symmetric simple exclusion processes with L=&ρmacr;N ^d particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N ^{− d} [∑ ^L ₁δ _{xi (·)}] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides the initial profile, only such canonical objects as bulk and self diffusion coefficients. Received: 7 September 1997 / Revised version: 15 May 1998     

Limit Theorems for Supercritical Branching Random Fields

An infinite particle system in R^d is considered where the initial distribution is POISSON ian and each initial particle gives rise to a supercritical age-dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space-time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS -MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWN ian motion and having exponentially distributed lifetime.     

On estimating the derivatives of symmetric diffusions in stationary random environment,with applications to ∇<Emphasis Type="Italic">ϕ</Emphasis> interface model

We consider diffusions on ℝ^d or random walks on ℤ^d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and H?lder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇ϕ interface model.     

Branching Random Walks in Space–Time Random Environment: Survival Probability,Global and Local Growth Rates

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.     

Multi-scale Clustering for a Non-Markovian Spatial Branching Process

Consider a system of particles which move in R^d according to a symmetric α-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+β. In case of the critical dimension d=α/β the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+β. Our result generalizes the case α=2 of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting. Supported in part by the DFG. Supported in part by the grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

Branching random walks and contact processes on homogeneous trees

Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d ^{− n/2} , while if there is global extinction, this probability is at most d ⁻ⁿ . Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper. Received: 26 April 1996/In revised form: 20 June 1996     

On symmetric random walks with random conductances on ℤ d

We study models of continuous time, symmetric, ℤ^d-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0 and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Localization transition of <Emphasis Type="Italic">d</Emphasis>-friendly walkers

 Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S ^k _j } _j≥0 , k=1,2,…,d by introducing ``non-crossing condition': and ``reward for collisions' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (β _m )≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d−1)-dimensional space for the parameters . Received: 13 June 2002 / Revised version: 24 August 2002 Published online: 28 March 2003 Mathematics Subject Classification (2000): 82B41, 82B26, 82D60, 60G50 Key words or phrases: Random walks – Random surfaces – Lattice animals – Phase transitions – Polymers – Random walks     

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

Let x and y be points chosen uniformly at random from ${\mathbb {Z}_n^4}$ , the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n ²(log n)^1/6, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on ${\mathbb {Z}_n^4}$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d ≥ 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.     

A Random Walk with Collapsing Bonds and Its Scaling Limit

Abstract

We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special cases. With minor changes the same argument can be used to prove the scaling limit of the corresponding walk in ? ^d .     

Equivalence of renewal sequences and isomorphism of random walks

We show that centred aperiodic random walks on ℤ ^d whose jump random variables are inL ²√log⁺ L have equivalent renewal sequences. An isomorphism theorem is deduced. Research was done while the author was visiting the Centre de Physique Theorique, Luminy-Marseille, France. Research supported by NSF Grant DMS 91-00725.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

Weak interaction limits for one-dimensional random polymers

 In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces, and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach followed in our earlier work, which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations. We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example (1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and find the scaling as the width tends to infinity. Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60 Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality     

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of sites, where ω _d is the volume of the unit ball in , we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(r ^α ) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr ² particles, we show that the inradius is at least , and the outradius is at most . This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig. Yuval Peres is partially supported by NSF grant DMS-0605166.     

On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets

We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ ^ds/2 as λ→∞ where we can explicitly compute the spectral dimension d _s . Moreover the limit N(λ) λ ^−ds/2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal. Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000     

Hypergroup deformations and Markov chains

Persi Diaconis and Phil Hanlon in their interesting paper⁽⁴⁾ give the rates of convergence of some Metropolis Markov chains on the cubeZ ^d (2). Markov chains on finite groups that are actually random walks are easier to analyze because the machinery of harmonic analysis is available. Unfortunately, Metropolis Markov chains are, in general, not random walks on group structure. In attempting to understand Diaconis and Hanlon's work, the authors were led to the idea of a hypergroup deformation of a finite groupG, i.e., a continuous family of hypergroups whose underlying space isG and whose structure is naturally related to that ofG. Such a deformation is provided forZ ^d (2), and it is shown that the Metropolis Markov chains studied by Diaconis and Hanlon can be viewed as random walks on the deformation. A direct application of the Diaconis-Shahshahani Upper Bound Lemma, which applies to random walks on hypergroups, is used to obtain the rate of convergence of the Metropolis chains starting at any point. When the Markov chains start at 0, a result in Diaconis and Hanlon⁽⁴⁾ is obtained with exactly the same rate of convergence. These results are extended toZ ^d (3).Research supported in part by the Office of Research and Sponsored Programs, University of Oregon.