On the concentration and the convergence rate with a moment condition in first passage percolation

TL;DR

This paper studies the concentration and convergence rate of passage times in first passage percolation on a lattice, focusing on moment conditions and employing a martingale approach under certain tail assumptions.

Contribution

It introduces a new analysis of concentration and convergence rates in first passage percolation using a martingale structure under moment conditions, extending previous exponential tail results.

Findings

Established concentration bounds under moment conditions

Derived convergence rate estimates for expected passage times

Utilized a novel martingale approach for analysis

Abstract

We consider the first passage percolation model on the ${\bf Z}^d$ lattice. In this model, we assign independently to each edge $e$ a non-negative passage time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage time from the origin to $(n,0,..., 0)$. Under the exponential tail assumption, Kesten (1993) and Talagrand (1995) investigated the concentration of $a_{0,n}$ from its mean using different methods. With this concentration and the exponential tail assumption, Alexander gave an estimate for the convergence rate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for $a_{0,n}$ using a special martingale structure.