Behavior of the distance exponent for \frac1|x-y|^2d long-range percolation

2022/08/09 by Bäumler, Johannes
#05C12 #60K35 #82B27 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2208.04793

Abstract

We study independent long-range percolation on ℤ^d where the vertices u and v are connected with probability asymptotic to \fracβ‖u-v‖^2d for ‖u-v‖_∞≥ 2 and with probability 1 for ‖u-v‖_∞=1, where β≥ 0 is a parameter. It is proven in [5] that there exists an exponent θ=θ(d,β) ∈ (0,1] such that the graph distance between the origin 0 and x ∈ ℤ^d scales like ‖x‖^θ. We prove that this exponent θ(d,β) is continuous and strictly decreasing as a function in β. Furthermore, we show that θ(d,β)=1-β+o(β) for small β in dimension d=1.

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