The polynomial growth of effective resistances in one-dimensional critical long-range percolation

2025/04/30 by Ding, Jian, Fan, Zherui, Huang, Lu-Jing · 1 citation
#60K35 #82B27 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.21378

Abstract

We study the critical long-range percolation on ℤ, where an edge connects i and j independently with probability 1-exp\-β∫_iⁱ⁺¹∫_j^j+1|u-v|^-2\rm d u\rm d v\ for |i-j|>1 for some fixed β>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^δ(β) for some δ(β)∈ (0,1).

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• Spectral dimensions for one-dimensional critical long-range percolation

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