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Entropy reduction in Euclidean first-passage percolation

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 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 $μn \leq \mathbb{E}T_n \leq μn + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy between the expected passage time and its deterministic approximation $μ= \lim_n \frac{\mathbb{E}T_n}{n}$. In this paper, we introduce an inductive entropy reduction technique that gives the stronger upper bound $\mathbb{E}T_n \leq μn + C_kψ(n) \log^{(k)}n$, where $ψ(n)$ is a general scale of concentration and $\log^{(k)}$ is the $k$-th iterate of $\log$. This gives evidence that the inequality $\mathbb{E}T_n - μn \leq C\sqrt{\mathrm{Var}~T_n}$ may hold.