First Passage Percolation with nonidentical passage times

Ghurumuruhan Ganesan

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arXiv · 1409.2602

First Passage Percolation with nonidentical passage times

Ghurumuruhan Ganesan

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Abstract

In this paper we consider first passage percolation on the square lattice $\mathbb{Z}^d$ with passage times that are independent and have bounded $p^{th}$ moment for some $p > 6(1+d),$ but not necessarily identically distributed. For integer $n \geq 1,$ let $T(0,n)$ be the minimum time needed to reach the point $(n,\mathbf{0})$ from the origin. We prove that $\frac{1}{n}\left(T(0,n) - \mathbb{E}T(0,n)\right)$ converges to zero in $L^2$ and use a subsequence argument to obtain almost sure convergence. As a corollary, for i.i.d. passage times, we also obtain the usual almost sure convergence of $\frac{T(0,n)}{n}$ to a constant $\mu.$

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Ghurumuruhan Ganesan. 2014-09-09. First Passage Percolation with nonidentical passage times. https://arxiv.org/abs/1409.2602

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