Subdiffusivity of random walk on the 2D invasion percolation cluster

TL;DR

This paper establishes almost sure lower bounds for the exit time of random walks on 2D invasion percolation clusters, demonstrating subdiffusive behavior and connecting it to percolation exponents.

Contribution

It provides the first almost sure subdiffusivity bounds for random walks on invasion percolation clusters, extending Kesten's theorem and explicitly relating bounds to percolation arm exponents.

Findings

Proves almost sure lower bounds for exit times on invasion percolation clusters.

Connects subdiffusivity to percolation arm exponents.

Provides explicit bounds involving the backbone exponent on the hexagonal lattice.

Abstract

We derive "quenched" subdiffusive lower bounds for the exit time tau(n) from a box of size n for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analog of H. Kesten's subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on the instrinsic distance in these graphs and general inequalities for reversible Markov chains. In the second part of the paper, we present a sharpening of Kesten's original argument, leading to an explicit almost sure lower bound for tau(n) in terms of percolation arm exponents. The methods give tau(n) \geq n^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distance and (assuming the exact value of the backbone exponent) kappa can be taken to be 17/384 on the hexagonal lattice.