The effective resistance and random walk in one-dimensional critical long-range percolation

Speaker: 范哲睿（北京大学）

Title: The effective resistance and random walk in one-dimensional critical long-range percolation

Inviter: 随机分析研究中心

Time & Venue: 2025年11月7日16:00--17:00 南楼613

Abstract: We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}\d u\d v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{\delta(\beta)}$ for some $\delta(\beta)\in (0,1)$. Finally, we will consider the heat kernel estimates of the random walk on this model. The talk is based on joint works with Jian Ding and Lu-Jing Huang.