On The Time Constant for Last Passage Percolation on Complete Graph

TL;DR

This paper investigates the asymptotic behavior of the maximum passage time in last passage percolation on complete graphs, establishing convergence, deviation decay rates, and bounds depending on the finiteness of the time constant.

Contribution

It proves the convergence of the scaled maximum passage time to the time constant and characterizes the decay of deviation probabilities, providing new insights into the model's probabilistic structure.

Findings

$W_n/n$ converges to the time constant $$

Deviation probability decays as $e^{-(n^2)}$ when $<$

Bounds for $W_n/n$ are established when $=$

Abstract

This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passage times of edges. Denote by $W_n$ the largest passage time among all self-avoiding paths from 1 to $n$. First, it is proved that $W_n/n$ converges to constant $\mu$, where $\mu$ is called the time constant and coincides with the essential supremum of $X_e$. Second, when $\mu<\infty$, it is proved that the deviation probability $P(W_n/n\leq \mu-x)$ decays as fast as $e^{-\Theta(n^2)}$, and as a corollary, an upper bound for the variance of $W_n$ is obtained. Finally, when $\mu=\infty$, lower and upper bounds for $W_n/n$ are given.