Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation

Abstract

Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[−\infty, 1$] with essential supremum equal to 1 (a charge of $−\infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F = p\delta_1 + (1 − p)\delta_{-\infty}$, corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call Max Growth System (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.