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May 2021 A shape theorem for the orthant model
Mark Holmes, Thomas S. Salisbury
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Ann. Probab. 49(3): 1237-1256 (May 2021). DOI: 10.1214/20-AOP1476

Abstract

We study a particular model of a random medium, called the orthant model, in general dimensions d2. Each site xZd independently has arrows pointing to its positive neighbours x+ei, i=1,,d with probability p and, otherwise, to its negative neighbours xei, i=1,,d (with probability 1p). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary which is a key requirement of the subadditive ergodic theorem.

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Mark Holmes. Thomas S. Salisbury. "A shape theorem for the orthant model." Ann. Probab. 49 (3) 1237 - 1256, May 2021. https://doi.org/10.1214/20-AOP1476

Information

Received: 1 November 2019; Revised: 1 June 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1476

Subjects:
Primary: 60K35

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 3 • May 2021
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