Abstract
We consider one-dependent random walks on ${\mathbb{Z} }^{d}$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function $\kappa $ of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that $\kappa $ coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges.
Highlights
We consider one-dependent random walks on Zd in random hypergeometric environment for d ≥ 3
The special case of random walks in random Dirichlet environment (RWDE), [ES06], where the environment is i.i.d. at each site and distributed according to a Dirichlet law, shows remarkable simplifications, while keeping the main phenomenological behavior as the general model. For this special choice of distribution, a key property of “statistical invariance by time reversing” makes it possible to prove transience in dimension d ≥ 3 [Sab11], existence of an invariant measure viewed from the particle absolutely continuous with respect to the static law, and equivalence between directional transience and ballisticity in dimension d ≥ 3 [ST11, Sab[13], Bou[13], ST17]
The aim of this paper is to give a generalization of this model and of these results to a class of one-dependent random walks in random environment, based on some hypergeometric distributions
Summary
Call functions of the following form hypergeometric functions: EJP 25 (2020), paper 33. The integral is computed according to the Lebesgue measure on the simplex n−1 du = du1 · · · dun−1 so that un = 1 − ui. When (Zj,i) has strictly positive coefficients, i=1 we have for all (Z · u)j ≥ z, with z = mini,j (Zj,i), so that the integral (1.2) is finite. These functions are classical generalized hypergeometric functions, see e.g. These functions are classical generalized hypergeometric functions, see e.g. [AKKI11, Section 3.7.4.]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
