Unions of random walk and percolation on infinite graphs

TL;DR

This paper studies the asymptotic behavior of a complex process combining random walks and percolation on infinite graphs, establishing laws of large numbers and variance estimates across different graph structures.

Contribution

It introduces a novel analysis of the superposition of random walk traces and percolation, providing new asymptotic results and variance estimates for various graph types.

Findings

Law of large numbers on vertex-transitive transient graphs

Similar behavior on finitely modified vertex-transitive graphs

Almost sure fluctuations on certain non-vertex-transitive graphs

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.