Critical first-passage percolation starting on the boundary

2016/12/06 by Jianping Jiang, Jiang, Jianping, Chang-Long Yao +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #60K35 #82B43 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1612.01803

openalex publication_date 2016/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider first-passage percolation on the two-dimensional triangular lattice T. Each site v\inT is assigned independently a passage time of either 0 or 1 with probability 1/2. Denote by B⁺(0,n) the upper half-disk with radius n centered at 0, and by c_n⁺ the first-passage time in B⁺(0,n) from 0 to the half-circular boundary of B⁺(0,n). We prove lim_n→∞(c_n⁺)/(log n)=(√(3))/(2π)~ a.s.,~lim_n→∞(E c_n⁺)/(log n)=(√(3))/(2π),~lim_n→∞(Var(c_n⁺))/(log n)=\frac2√(3)π-(9)/(π²). These results enable us to prove limit theorems with explicit constants for any first-passage time between boundary points of Jordan domains. In particular, we find the explicit limit theorems for the cylinder point to point and cylinder point to line first-passage times.

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