Multiple phase transitions in long-range first-passage percolation on square lattices

2013/09/23 by Shirshendu Chatterjee, Chatterjee, Shirshendu, Partha S. Dey +1 · 2 citations
Mathematics · Physics and Astronomy · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.1309.5757

Final version, 46 pages; 6 figures. To appear in CPAM

openalex publication_date 2013/09/23 · arxiv created 2015/03/03 · arxiv updated 2015/03/04 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider a model of long-range first-passage percolation on the d dimensional square lattice Z^d in which any two distinct vertices x, y ∈ Z^d are connected by an edge having exponentially distributed passage time with mean ||x-y||^α+o(1), where α>0 is a fixed parameter and ||⋅|| is the ℓ₁-norm on Z^d. We analyze the asymptotic growth rate of the set B_t, which consists of all x ∈ Z^d such that the first-passage time between the origin 0 and x is at most t, as t→∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α<d, (ii) stretched exponential growth for α∈ (d,2d), (iii) superlinear growth for α∈ (2d,2d+1) and finally (iv) linear growth for α>2d+1 like the nearest-neighbor first-passage percolation model corresponding to α=∞.

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