Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

2024/05/06 by Jian Ding, Ding, Jian, Zherui Fan +3 · 1 citation
Mathematics · Physics and Astronomy · #60K35 #82B27 #82B43 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2405.03460

openalex publication_date 2024/05/06 · openalex created_date 2024/05/10 · openalex updated_date 2026/07/28

Abstract

In this work, we study the critical long-range percolation on ℤ, where an edge connects i and j independently with probability 1-exp\-β|i-j|^-2\ for some fixed β>0. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 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 β>0 and thus rules out a potential phase transition (around β= 1) which seemed to be a reasonable possibility.

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