Convergence in First Passage Percolation with nonidentical passage times

2017/04/03 by Ghurumuruhan Ganesan, Ganesan, Ghurumuruhan
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1704.00459

arxiv created 2017/04/03 · arxiv updated 2017/04/04

Abstract

In this paper we consider first passage percolation on the square lattice \(ℤ^d\) with edge passage times that are independent and have uniformly bounded second moment, but not necessarily identically distributed. For integer \(n ≥ 1,\) let \(T_n\) be the minimum passage time between the origin and the point \((n,0,…,0).\) We prove that \((1)/(n)(T_n-𝔼T_n)\) converges to zero almost surely and in \(L²\) as \(n~→~∞.\) The convergence is nontrivial in the sense that \((T_n)/(n)\) is asymptotically bounded away from zero and infinity almost surely. We first define a truncated version \(T⁽ⁿ⁾_n\) that is asymptotically equivalent to~\(T_n.\) We then use a finite box modification of the martingale method of Kesten~(1993) to estimate the variance of \(T⁽ⁿ⁾_n.\) Finally, we use a subsequence argument to obtain almost sure convergence for \((1)/(n)(T⁽ⁿ⁾_n - 𝔼T⁽ⁿ⁾_n).\) The corresponding result for \(T_n\) is then obtained using asymptotic equivalence of \(T_n\) and \(T⁽ⁿ⁾_n.\) For identically distributed passage times, our method alternately obtains almost sure convergence of~\((T_n)/(n)\) to a positive constant~\(μ_F,\) without invoking the subadditive ergodic theorem.

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