2020/10/15 by Damron, Michael, Christopher Janjigian, Hanson, Jack +5
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2010.08072
openalex publication_date 2020/10/15 · openalex created_date 2020/10/22 · openalex updated_date 2026/07/28
In first-passage percolation, we assign i.i.d.~nonnegative weights (te) to the nearest-neighbor edges of ℤd and study the induced pseudometric T = T(x,y). In this paper, we focus on geodesics, or optimal paths for T, and estimate the empirical distribution of weights along them. We prove an upper bound for the expected number of edges with weight ≥ M in the union of all geodesics from 0 to x of the form q(M) ℙ(te ≥ M)|x|, where q(M) ≤ e-cM. This shows that the tail of the expected empirical distribution along a geodesic is lighter than that of the original weight distribution by an exponential factor. We also give a lower bound for the expected minimal number of edges with weight ≥ M in any geodesic from 0 to x in terms of ℙ(te ≥ M) and ℙ(te ∈ [M,2M]). For example, these two imply that if te has a power law tail of the form ℙ(te ≥ M) ∼ M-α, then the tail of the expected empirical distribution asymptotically lies between e-CM log M and e-cM. We also provide estimates for the expected number of edges in a geodesic with weight in a set A for (a) arbitrary A, (b) A an interval separated from the infimum of the support of te and (c) A=[0,a] for some a ≥ 0.