Скачать или смотреть Yuval Peres: Random walk on dynamical percolation: separating critical and supercritial...

Yuval Peres: Random walk on dynamical percolation: separating critical and supercritial regimes #ICBS2024
Let $G$ be an infinite regular tree $T$ or a $d$-dimensional lattice. We study the random walk on dynamical percolation in $G$, where each edge can either be open or closed, refreshing its status at rate $\mu\in (0,1/e]$. The random walk traverses $G$ along open edges, attempting jumps at rate $1$. For the critical regime $p=p_c$ on $T$, we prove an upper bound on the speed of the walk that is a power of $\mu$; on high-dimensional and two-dimensional lattices, the mean squared displacement of the walk at time $t$ is at most $O(t \mu^{\alpha_d} )$, where $\alpha_d$ depends on the one-arm exponent for critical percolation in dimension $d$. For the supercritical regime $p(lg)p_c$, we prove that the speed on $T$ is of order 1, while for the subcritical regime $p