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Almost invariance of distributions for random walks on groups

Abstract

We study the neighborhoods of a typical point \(Z_n\) visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to \(\mu ^{*n}(Z_n)\). If such neighborhood contains a ball of radius \(C \sqrt{n}\), we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of \(\mathbb {Z}\) with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property \(H_\mathrm{FD}\). We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of \(\mathbb {Z}^2\) with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

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Acknowledgements

I would like to thank Vadim Kaimanovich for helpful discussions and comments on the preliminary version of this paper and Balint Toth for turning my attention to the result of [4]. I am grateful to the referee for helpful remarks that significantly improved the exposition of the paper.

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Correspondence to Anna Erschler.

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The work of the author is supported by the ERC grant GroIsRan and by the ANR grant MALIN.

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Erschler, A. Almost invariance of distributions for random walks on groups. Probab. Theory Relat. Fields 174, 445–476 (2019). https://doi.org/10.1007/s00440-019-00915-3

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  • DOI: https://doi.org/10.1007/s00440-019-00915-3

Mathematics Subject Classification

  • 60B15
  • 60G50
  • 05C81
  • 20F65
  • 20E22