Collection Of Random Walks Research Articles

Avoidance Coupling

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.

A random energy model for size dependence: recurrence vs. transience

We investigate the size dependence of disordered spin models having an infinite number of Gibbs measures in the framework of a simplified 'random energy model for size dependence'. We introduce two versions (involving either independent random walks or branching processes), that can be seen as generalizations of Derrida's random energy model. Our model is shown to exhibit a recurrence/transience transition for Gibbs measures (meaning: a.s. existence/nonexistence of subsequences of volumes converging to a given random infinite volume state), depending on the growthrate of a function MN describing the 'number of extremal Gibbs states that can be observed in a finite volume N'. We investigate the model in detail in the 'critical regime' MN ~ (log N)^p (with critical point p = 1). We obtain the a.s. large volume asymptotics of the relative weights for finding a particular state and we compute the set of a.s. cluster points of the corresponding occupation times (corresponding to the 'empirical metastate'). In the course of the proof we obtain Laws of the Iterated Logarithm for the pair XνN, YνN := supµ=1,2,...,MN;µ≠ν XµN where (XµN)µ=1,...,MN;N=1,2,... is either a branching random walk or a collection of random walks, independent over µ.

Random walks with negative drift conditioned to stay positive

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ < 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.