Abstract: In Dynamical Percolation each edge is open with probability p, refreshing its status at rate \mu>0. This process was introduced in the 1990s by Haggstrom, Steif and the speaker, motivated by a question of Malliavin. Remarkable results on exceptional times in two dimensions were obtained by Schramm, Steif, Garban and Pete.
We study random walk on dynamical percolation in the lattice Z^d, where the walk moves along open edges at rate 1. Let p_c=p_c(d) denote the critical value for static percolation. In the critical regime p=p_c, we prove that if d=2 or d>10, then the mean squared displacement is O(t \mu^a) where a=a(d)>0. For p>p_c, we prove that the mean squared displacement is of order t, uniformly in 0<\mu<1, refining earlier results obtained with Sousi and Steif. (For p<p_c and \mu<1, it is known that the mean squared displacement is of order t \mu.) We will show simulations to illustrate the process.
(Joint work with Chenlin Gu, Jianping Jiang, Zhan Shi, Hao Wu and Fan Yang.)
About the Speaker: Yuval Peres obtained his PhD in 1990 from the Hebrew University, Jerusalem. He was a postdoctoral fellow at Stanford and Yale, and was then a Professor of Mathematics and Statistics in Jerusalem and in Berkeley. Later, he was a Principal researcher at Microsoft. In 2023, he joined Beijing Institute of Mathematical Sciences and Applications. He has published more than 350 papers in most areas of probability theory, including random walks, Brownian motion, percolation, and random graphs. He has co-authored books on Markov chains, probability on graphs, game theory and Brownian motion, which can be found at https://www.yuval-peres-books.com. His presentations are available at https://yuval-peres-presentations.com. He is a recipient of the Rollo Davidson prize and the Loeve prize. He has mentored 21 PhD students including Elchanan Mossel (MIT, AMS fellow), Jian Ding (PKU, ICCM gold medal and Rollo Davidson prize), Balint Virag and Gabor Pete (Rollo Davidson prize). He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, at the 2008 European congress of Math, and at the 2017 Math Congress of the Americas. In 2016, he was elected to the US National Academy of Science.
