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

YMSC Probability Seminar

Organizers：

吴昊，杨帆，姜建平，顾陈琳，李文博

Speaker：

Zherui Fan 范哲睿 (PKU)

Time：

Thur., 16:00-17:00, Nov. 20, 2025

Venue：

C548, Shuangqing Complex Building A

Title:

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

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.