The dimension of the range of a transient random walk
Nicos Georgiou, Davar Khoshnevisan, Kunwoo Kim, Alex D. Ramos
TL;DR
This paper derives formulas for the Minkowski and Hausdorff dimensions of the range of any transient random walk in Z^d, addressing a problem posed by Barlow and Taylor in 1991.
Contribution
It provides the first general formulas for the macroscopic dimensions of transient random walk ranges in Z^d, solving a longstanding open problem.
Findings
Formulas for Minkowski and Hausdorff dimensions of the walk's range
Addresses and solves Barlow and Taylor's 1991 problem
Applicable to arbitrary transient walks in Z^d
Abstract
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
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Taxonomy
TopicsStochastic processes and statistical mechanics · Diffusion and Search Dynamics · Mathematical Dynamics and Fractals