Spectral dimensions for one-dimensional critical long-range percolation

TL;DR

This paper determines the spectral dimensions of a one-dimensional critical long-range percolation model, showing they depend on the resistance exponent, thus resolving an open question in the field.

Contribution

It establishes the spectral dimension for critical long-range percolation on $ ext{Z}$, linking it explicitly to the resistance exponent, and addresses an open problem from prior research.

Findings

Spectral dimension is 2/(1+δ) for the model.

Results hold for both quenched and annealed cases.

Addresses an open question from previous work.

Abstract

Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $\beta>0$ 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 $2/(1+\delta)$, where $\delta\in (0,1)$ 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].