The Annals of Probability

Random Dirichlet environment viewed from the particle in dimension $d\ge3$

Christophe Sabot

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Abstract

We consider random walks in random Dirichlet environment (RWDE), which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb{Z}}^{d}$, RWDE are parameterized by a $2d$-tuple of positive reals called weights. In this paper, we characterize for $d\ge3$ the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle. We can deduce from this result and from [Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8] a complete description of the ballistic regime for $d\ge3$.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 722-743.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750940

Digital Object Identifier
doi:10.1214/11-AOP699

Mathematical Reviews number (MathSciNet)
MR3077524

Zentralblatt MATH identifier
1269.60077

Subjects
Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random walk in random environment Dirichlet distribution reinforced random walks invariant measure viewed from the particle

Citation

Sabot, Christophe. Random Dirichlet environment viewed from the particle in dimension $d\ge3$. Ann. Probab. 41 (2013), no. 2, 722--743. doi:10.1214/11-AOP699. https://projecteuclid.org/euclid.aop/1362750940


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