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Orenshtein, Tal; Sabot, Christophe

Random walks in random hypergeometric environment. (English) Zbl 1441.60085

Electron. J. Probab. 25, Paper No. 33, 21 p. (2020).
Summary: We consider one-dependent random walks on \(\mathbb{Z}^d\) in random hypergeometric environment for \(d\geq 3\). 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 [the second author, Probab. Theory Relat. Fields 151, No. 1–2, 297–317 (2011; Zbl 1231.60121)] and [the second author, Ann. Probab. 41, No. 2, 722–743 (2013; Zbl 1269.60077)] 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments

Keywords:

random walks in random environment; point of view of particle; hypergeometric functions; hypergeometric environments; Dirichlet environments; reversibility; one-dependent Markov chains

Citations:

Zbl 1231.60121; Zbl 1269.60077
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\textit{T. Orenshtein} and \textit{C. Sabot}, Electron. J. Probab. 25, Paper No. 33, 21 p. (2020; Zbl 1441.60085)
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