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arXiv · 2406.01580

Secret sharing on the Poisson-Furstenberg boundary I

Andrei Alpeev
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Abstract

Recently, Vadim Kaimanovich presented a particular example of a measure on a product of two standard lamplighter groups such that the Poisson boundary of the induced random walk is non-trivial, but the boundary on the marginals is trivial. This was surprising since such behavior is not possible for measures of finite entropy. As we show in this paper, this secret-sharing phenomenon is possible precisely for pairs of amenable groups with non-trivial ICC-factors.

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Andrei Alpeev. 2024-06-03. Secret sharing on the Poisson-Furstenberg boundary I. https://arxiv.org/abs/2406.01580

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