Speaker: 黄璐静（福建师范大学）
Title: The effective resistance in one-dimensional critical long-range percolation
Inviter: 随机分析研究中心
Time & Venue: 2025 年11月7日15:00–16:00 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 have a polynomial lower bound in n. Our bound holds for all $\beta>0$ and thus rules out a potential phase transition (around $\beta=1$) which seemed to be a reasonable possibility. Finally, I will introduce some other new progresses in this topic. This talk is based on joint works with Jian Ding and Zherui Fan.
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