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Entropy reduction in Euclidean first-passage percolation. (English) Zbl 1354.60116
Summary: The Euclidean first-passage percolation (FPP) model of C. D. Howard and C. M. Newman [Probab. Theory Relat. Fields 108, No. 2, 153–170 (1997; Zbl 0883.60091)] 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 82B43 Percolation
Zbl 0883.60091
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