Chemical subdiffusivity of critical 2D percolation

TL;DR

This paper demonstrates that random walks on the incipient infinite cluster of 2D critical percolation are subdiffusive in the chemical distance, providing explicit bounds based on percolation exponents and extending results to stationary random graphs with polynomial volume growth.

Contribution

It establishes subdiffusivity of random walks in chemical distance on 2D critical percolation clusters and generalizes the result to certain stationary random graphs with explicit bounds.

Findings

Random walk on 2D critical percolation IIC is subdiffusive in chemical distance.

Explicit bounds on subdiffusivity are derived using percolation exponents.

Results extend to stationary random graphs with polynomial volume growth.

Abstract

We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multi-scale way of covering the graph so that "deep patches" have "thin backbones". Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents $\eta_2 > \eta_1 > 0$: For $d$-dimensional models, the mean chemical displacement after $T$ steps of random walk scales asymptotically slower than $T^{1/\beta}$, whenever \[ \beta < 2 + \frac{\eta_2-\eta_1}{d-\eta_1}\,. \] Using the conjectured values of $\eta_2 = \eta_1 + 1/4$ and $\eta_1 = 5/48$ for 2D lattices, the latter quantity is $2+12/91$.