Spectral dimensions for one-dimensional critical long-range percolation

2025/05/21 by Fan, Zherui, Huang, Lu-Jing
#60K35 #82B27 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.15037

Abstract

Consider 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|^-2d ud v\ for |i-j|>1 for some fixed β>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+δ), where δ∈ (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].

Citations

• The polynomial growth of effective resistances in one-dimensional critical long-range percolation
• Spectral dimension of simple random walk on a long-range percolation cluster
• Heat kernel estimates for strongly recurrent random walk on random media
• Random walk on the incipient infinite cluster for oriented percolation in high dimensions
• Random walks on supercritical percolation clusters

Related

• Spectral dimension of simple random walk on a long-range percolation cluster
• Critical long-range percolation II: Low effective dimension
• Power-law bounds for critical long-range percolation below the\n upper-critical dimension
• The polynomial growth of effective resistances in one-dimensional critical long-range percolation
• Critical long-range percolation I: High effective dimension