D\'eviations mod\'er\'ees de la distance chimique

TL;DR

This paper investigates the fluctuations of the chemical distance in Bernoulli percolation, providing improved estimates on convergence to the asymptotic shape using concentration inequalities and subadditive function theory.

Contribution

It establishes moderate deviations for the chemical distance, enhancing understanding of its fluctuations and asymptotic behavior in Bernoulli percolation.

Findings

Derived new bounds for the fluctuations of the chemical distance

Improved understanding of convergence to the asymptotic shape

Applied concentration inequalities and subadditive function approximation

Abstract

In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.