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Mathematics > Probability
Title: A Notion of Dimension based on Probability on Groups
(Submitted on 26 Apr 2024)
Abstract: We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $\mu$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $\mu$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups.
Submission history
From: Agelos Georgakopoulos [view email][v1] Fri, 26 Apr 2024 09:32:32 GMT (101kb,D)
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