Entropy reduction in Euclidean first-passage percolation

Abstract

The Euclidean first-passage percolation (FPP) model of Howard and Newman is a rotationally invariant model of FPP which is built on a graph whose vertices are the points of homogeneous Poisson point process. It was shown by Howard-Newman that one has (stretched) exponential concentration of the passage time $T_n$ from $0$ to $n\mathbf{e} _1$ about its mean on scale $\sqrt{n} $, and this was used to show the bound $\mu n \leq \mathbb{E} T_n \leq \mu n + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy between the expected passage time and its deterministic approximation $\mu = \lim _n \frac{\mathbb {E}T_n} {n}$. In this paper, we introduce an inductive entropy reduction technique that gives the stronger upper bound $\mathbb{E} T_n \leq \mu n + C_k\psi (n) \log ^{(k)}n$, where $\psi (n)$ is a general scale of concentration and $\log ^{(k)}$ is the $k$-th iterate of $\log $. This gives evidence that the inequality $\mathbb{E} T_n - \mu n \leq C\sqrt{\mathrm {Var}~T_n} $ may hold.