Rate of growth of a transient cookie random walk
نویسندگان
چکیده
We consider a one-dimensional transient cookie random walk. It is known from a previous paper [3] that a cookie random walk (Xn) has positive or zero speed according to some positive parameter α > 1 or ≤ 1. In this article, we give the exact rate of growth of (Xn) in the zero speed regime, namely: for 0 < α < 1, Xn/n α+1 2 converges in law to a Mittag-Leffler distribution whereas for α = 1, Xn(log n)/n converges in probability to some positive constant.
منابع مشابه
Large deviations and slowdown asymptotics for one - dimensional excited random walks ∗
We study the large deviations of excited random walks on Z. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed v0, then the large deviation rate function for the position of the excited random walk is zero on the inte...
متن کاملThe excited random walk in one dimension
We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q = 1−p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scal...
متن کاملRandom Walks on Directed Covers of Graphs
Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates do not longer coincide in general. Furthermore, the behaviour of random walks on directed covers of infi...
متن کاملExcited Random Walk in Two Dimensions Has Linear Speed
Imagine a cookie placed on every vertex of an infinite d-dimensional grid. A random walker on this grid behaves as follows upon encountering a cookie: he consumes it and then performs a random step with an ǫ-drift to the right, namely the probability to make a right move is 1 2d + ǫ, the probability for a left move is 1 2d − ǫ and the probability for all other moves is 1 2d . When encountering ...
متن کاملRange of a Transient 2d-Random Walk
We study the range of a planar random walk on a randomly oriented lattice, already known to be transient. We prove that the expectation of the range grows linearly, in both the quenched (for a.e. orientation) and annealed (”averaged”) cases. We also express the rate of growth in terms of the quenched Green function and eventually prove a weak law of large numbers in the (non-Markovian) annealed...
متن کامل