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Random Walk on a Perturbation of the Infinitely-Fast Mixing Interchange Process

Salvi, Michele; Simenhaus, François (2018), Random Walk on a Perturbation of the Infinitely-Fast Mixing Interchange Process, Journal of Statistical Physics, 171, 4, p. 656-678. 10.1007/s10955-018-2015-z

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Type
Article accepté pour publication ou publié
Date
2018
Journal name
Journal of Statistical Physics
Volume
171
Number
4
Publisher
Springer
Pages
656-678
Publication identifier
10.1007/s10955-018-2015-z
Metadata
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Author(s)
Salvi, Michele
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Simenhaus, François
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider a random walk in dimension d≥1 in a dynamic random environment evolving as an interchange process with rate γ>0. We prove that, if we choose γ large enough, almost surely the empirical velocity of the walker Xt/t eventually lies in an arbitrary small ball around the annealed drift. This statement is thus a perturbation of the case γ=+∞ where the environment is refreshed between each step of the walker. We extend three-way part of the results of Huveneers and Simenhaus (Electron J Probab 20(105):42, 2015), where the environment was given by the 1-dimensional exclusion process: (i) We deal with any dimension d≥1; (ii) We treat the much more general interchange process, where each particle carries a transition vector chosen according to an arbitrary law μ; (iii) We show that Xt/t is not only in the same direction of the annealed drift, but that it is also close to it.
Subjects / Keywords
Random walk; Dynamic random environment; Interchange process; Limit theorems; Renormalisation

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