Greedy lattice paths with general weights

TL;DR

This paper studies the asymptotic behavior of maximum weights of self-avoiding lattice paths with general weights, establishing convergence results under certain integrability conditions, and relates the model to percolation and branching random walks.

Contribution

It introduces a unified analysis of lattice path weights with general distributions, bridging first and last passage percolation models, and proves convergence of scaled maximum weights.

Findings

Proves $M_n/n$ converges in $L^1$ to a constant $M$.

Establishes almost sure convergence of $M_n/n$ to $M$ under stronger conditions.

Connects the model to percolation and branching random walk theories.

Abstract

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(\pi)=\sum_{v\in\pi}X_v$ be the weight of a self-avoiding lattice path $\pi$. Let \[M_n=\max\{S(\pi):\pi\text{ has length }n\text{ and starts from the origin}\}.\] 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 $\exists\alpha>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ 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 $L^{1}$ 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.