Two-sided infinite-bin models and analyticity for Barak-Erd\H{o}s graphs

TL;DR

This paper constructs a two-sided stationary infinite-bin model for any distribution, derives a new series formula for its front speed, and proves the analyticity of the growth rate of the longest path in Barak-Erdős graphs.

Contribution

It introduces a method to create a two-sided stationary infinite-bin model for any distribution and establishes the analyticity of the growth rate in Barak-Erdős graphs.

Findings

A new series formula for the front speed of infinite-bin models.

Proof of the analyticity of the growth rate C(p) in Barak-Erdős graphs.

Construction of a two-sided stationary version of the infinite-bin model.

Abstract

In this article, we prove that for any probability distribution $\mu$ on $\mathbb{N}$ one can construct a two-sided stationary version of the infinite-bin model (an interacting particle system introduced by Foss and Konstantopoulos) with move distribution $\mu$. Using this result, we obtain a new formula for the speed of the front of infinite-bin models, as a series of positive terms. This implies that the growth rate $C(p)$ of the longest path in a Barak-Erd\H{o}s graph of parameter $p$ is analytic on $(0,1]$.