Spectral dimensions for one-dimensional critical long-range percolation

Abstract

Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp \{-{\hat{I}}^2{\int }_i^{i+1}{\int }_j^{j+1}|u-v{|}^{-2}dudv\}$ for $|i-j|>1$ for some fixed ${\hat{I}}^2>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+\hat{I}´)$, where $\hat{I}´\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5].

14 pages