Random Walk on Dynamical Percolation: Separating Critical and Supercritical Regimes

Explore a comprehensive lecture on random walks on dynamical percolation, focusing on separating critical and supercritical regimes. Delve into the study of random walks on infinite regular trees and d-dimensional lattices, where edges can be open or closed, refreshing their status at a specific rate. Examine the behavior of random walks traversing these structures along open edges, with attempted jumps occurring at a fixed rate. Investigate the critical regime on trees, uncovering an upper bound on the walk's speed expressed as a power of the refresh rate. Analyze the mean squared displacement of walks on high-dimensional and two-dimensional lattices at critical percolation, relating it to the one-arm exponent. Contrast these findings with the supercritical regime, where the speed on trees is shown to be of order 1. Gain insights into the mathematical techniques used to derive these results and their implications for understanding dynamical percolation processes.