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August 2019 Unions of random walk and percolation on infinite graphs
Kazuki Okamura
Braz. J. Probab. Stat. 33(3): 586-637 (August 2019). DOI: 10.1214/18-BJPS404

Abstract

We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple random walk on the same graph. We investigate asymptotics for the number of vertices of the enlargement of the trace of the walk until a fixed time, when the time tends to infinity. This process is more highly self-interacting than the range of random walk, which yields difficulties. We show a law of large numbers on vertex-transitive transient graphs. We compare the process on a vertex-transitive graph with the process on a finitely modified graph of the original vertex-transitive graph and show their behaviors are similar. We show that the process fluctuates almost surely on a certain non-vertex-transitive graph. On the two-dimensional integer lattice, by investigating the size of the boundary of the trace, we give an estimate for variances of the process implying a law of large numbers. We give an example of a graph with unbounded degrees on which the process behaves in a singular manner. As by-products, some results for the range and the boundary, which will be of independent interest, are obtained.

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Kazuki Okamura. "Unions of random walk and percolation on infinite graphs." Braz. J. Probab. Stat. 33 (3) 586 - 637, August 2019. https://doi.org/10.1214/18-BJPS404

Information

Received: 1 July 2017; Accepted: 1 May 2018; Published: August 2019
First available in Project Euclid: 10 June 2019

zbMATH: 07094819
MathSciNet: MR3960278
Digital Object Identifier: 10.1214/18-BJPS404

Rights: Copyright © 2019 Brazilian Statistical Association

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Vol.33 • No. 3 • August 2019
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