The one-arm exponent for mean-field long-range percolation

2014/11/11 by Hulshof, Tim
#60K35 (primary) #82B27 #82B43 (secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1411.3020

Abstract

Consider a long-range percolation model on ℤ^d where the probability that an edge \x,y\ ∈ ℤ^d × ℤ^d is open is proportional to ‖x-y‖₂^-d-α for some α>0 and where d > 3 min\2,α\. We prove that in this case the one-arm exponent equals min\4,α\/2. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter α when α=4.

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