Sublinear variance in Euclidean first-passage percolation

Megan Bernstein, Michael Damron and Torin Greenwood

Stochastic Processes and their Applications, 2020, vol. 130, issue 8, 5060-5099

Abstract: The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and ne1 obeys a diffusive upper bound: VarT(0,ne1)≤Cn, and in this paper we improve this inequality to Cn∕logn. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik–Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several “greedy lattice animal” arguments.

Keywords: First-passage percolation; Euclidean first-passage percolation; Sublinear variance; Entropy method; Bernoulli encoding; Logarithmic Sobolev inequality (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:130:y:2020:i:8:p:5060-5099

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DOI: 10.1016/j.spa.2020.02.011

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