On the time constant of high dimensional first passage percolation, revisited

2025/01/20 by Antonio Auffinger, Auffinger, Antonio, Si Tang +1 · 1 citation
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Complex Network Analysis Techniques

paper · pdf · doi:10.48550/arxiv.2501.11589

Abstract

In [2], it was claimed that the time constant μ_d(e₁) for the first-passage percolation model on \mathbb Z^d is μ_d(e₁) ∼ log d/(2ad) as d→ ∞, if the passage times (τ_e)_{e∈ \mathbb E^d} are i.i.d., with a common c.d.f. F satisfying |(F(x))/(x)-a| ≤ (C)/(|log x|) for some constants a, C and sufficiently small x. However, the proof of the upper bound, namely, Equation (2.1) in [2] \limsup_d→∞ \fracμ_d(e₁)adlog d ≤ (1)/(2) is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order o((log d/d)²) as d→ ∞ in the case of the when the edge weights are exponentially distributed.

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