Let (G,μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G,μ)-stationary space, with respect to the...

We investigate how the stationary distribution of a Markov chain changes when transitions from a single state are modified. In particular, adding a single directed edge to nearest neighbor random walk on a finite discrete torus in dimensions one, two, or three changes the stationary distribution linearly, logarithmically, or only locally. Related results are derived for birth and death chains a...

We first obtain by analogy with the continuous (differential) case the general solution of a discrete Riccati equation. Our results can be considered the discrete analog of Miel-nik's construction in supersymmetric quantum mechanics [J. Moreover, we establish the full equivalence of our discrete Riccati equation and a corresponding homogeneous second order discrete linear equation. We present a...

In this paper, we study throughput and delay scaling laws of wireless network with network coding under different mobility models. Specifically, we consider 2-hop and multi-hop schemes with n nodes and k original packets for each pair of source-destination. We study two ad hoc network mobility modelshybrid random walk models(HRWM) and discrete random direction models(DRDM). For the hybrid rando...

Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Abstract. We study a general transition operator, generated by a random walk on a graph X ; in particular we give necessary and sufficient condition on the matrix coefficient (1-step transition probablilities) to be a bounded operator from l(X) into itself. Moreover we characterize compact operators and we relate this property to the behaviour of the associated random walk. We give a necessary ...

In this lecture we provide two examples of problems where reasonable models are given by the deterministic continuum limit of the discrete random walk through which the problem is naturally defined. Once suitably defined, this continuum limit usually takes the form of a partial differential equation (hence deterministic). One interesting use of the continuum limit we will see in upcoming lectur...

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments wh...

We focus on two models of nearest-neighbour random walks on d-dimensional regular hyper-cubic lattices that are usually assumed to be identical—the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll–Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed ra...