Greedy Lattice Paths with General Weights

Abstract

Let {X_υ: υ ∈ ℤ^d} be i.i.d. random variables. Let $S(\pi) = \sum\nolimits_{\upsilon\in \pi} {{X_\upsilon}}$ be the weight of a self-avoiding lattice path π. Let

$${M_n} = \max\{ S(\pi):\,\,\pi\,{\text{has}}\,{\text{length}}\,n\,{\text{and}}\,{\text{starts}}\,{\text{from}}\,{\text{origin}}\}.$$

We are interested in the asymptotics of M_n as n → ∞. This model is closely related to the first passage percolation when the weights {X_υ: υ ∈ ℤ^d} are non-positive and it is closely related to the last passage percolation when the weights {X_υ, υ ∈ 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 $\exists \alpha > 0,\,E{(X_0^ + )^d}{({\log ^ + }X_0^ + )^{d + \alpha }} < + \,\infty$ 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 L¹ as n tends to infinity. And under the stronger assumptions that $\exists \alpha > 0,\,\,E{(X_0^ + )^d}{({\log ^ + }\,X_0^ + )^{d + \alpha }} < \, + \,\infty$ 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.