Greedy lattice paths with general weights

Abstract

Let $\{X_v:v\in {\mathbb{Z}}^d\}$ be i.i.d. random variables. Let $S(\ddot{I}€)={\sum }_{v\in \ddot{I}€}{X}_v$ be the weight of a self-avoiding lattice path $\ddot{I}€$. Let \[M_n=\max\{S(π):π\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to \infty$. This model is closely related to the first passage percolation when the weights $\{X_v:v\in {\mathbb{Z}}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in {\mathbb{Z}}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that $\exists \hat{I}\pm >0$, $E(X_0^{+}{)}^d({\log }^{+}{X}_0^{+}{)}^{d+\hat{I}\pm }<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^1$ as $n$ tends to infinity. And under the stronger assumptions that $\exists \hat{I}\pm >0$, $E(X_0^{+}{)}^d({\log }^{+}{X}_0^{+}{)}^{d+\hat{I}\pm }<+\infty$ and that $E[(X_0^{-}{)}^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity.