A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich (Ann. Math. 152:659–692, 2000). A geometrical method for constructing the strip as a subset of the lamplighter group ${\mathbb {Z}_{2}\wr \Gamma}$ starting with a “smaller” strip in the group Γ is developed. Then, this method is applied to several classes of base groups Γ: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.     

A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich (Ann. Math. 152:659–692, 2000). A geometrical method for constructing the strip as a subset of the lamplighter group \mathbb Z₂\wr G{\mathbb {Z}_{2}\wr \Gamma} starting with a “smaller” strip in the group Γ is developed. Then, this method is applied to several classes of base groups Γ: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.     

A note on random walks with absorbing barriers and sequential Monte Carlo methods

In this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.     

Persistent random walks in random environment

Summary Weak convergence of a class of functionals of PRWRE is proved. As a consequence CLT is obtained for the normed trajectory.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (Grant No. 476/82).     

The rate of escape for anisotropic random walks in a tree

Summary Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z _n on G or T which applies to the distance of Z _n from a fixed point, as well as to the distribution of the last R letters in Z _n . For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z _n with a finitely supported step distribution is derived and discussed.Partially supported by grant NSF MCS85-04315     

Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

Clifford-algebraic random walks on the hypercube

The n-dimensional hypercube is a simple graph on 2ⁿ vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing.     

Stratified random walks on the n-cube

In this paper we present a method for analyzing a general class of random walks on the n-cube (and certain subgraphs of it). These walks all have the property that the transition probabilities depend only on the level of the point at which the walk is. For these walks, we derive sharp bounds on their mixing rates, i.e., the number of steps required to guarantee that the resulting distribution is close to the (uniform) stationary distribution. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 199–222, 1997