Subdiffusivity of random walk on the 2D invasion percolation cluster

We derive quenched subdiffusive lower bounds for the exit time τ(n)τ(n) from a box of size nn for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analogue 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 intrinsic 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 τ(n)τ(n) in terms of percolation arm exponents. The methods give τ(n)≥n2+ϵ0+κτ(n)≥n2+ϵ0+κ, where ϵ0>0ϵ0>0 depends on the intrinsic distance and κκ can be taken to be 5384 on the hexagonal lattice.