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Article Dans Une Revue Probability Theory and Related Fields Année : 2022

Law of large numbers for the drift of the two-dimensional wreath product

Tianyi Zheng

Résumé

Abstract We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n -step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

Dates et versions

hal-03958744 , version 1 (26-01-2023)

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Anna Erschler, Tianyi Zheng. Law of large numbers for the drift of the two-dimensional wreath product. Probability Theory and Related Fields, 2022, 182 (3-4), pp.999-1033. ⟨10.1007/s00440-021-01098-6⟩. ⟨hal-03958744⟩
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