Chemical Subdiffusivity of Critical 2D Percolation

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 (Probab Theory Related Fields 73(3): 369–394 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 multiscale 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 $$\begin{aligned} \beta < 2 + \frac{\eta _2-\eta _1}{d-\eta _1}\,. \end{aligned}$$Using the conjectured values of \(\eta _2 = \eta _1 + 1/4\) and \(\eta _1 = 5/48\) for 2D lattices, the latter quantity is \(2+12/91\).