Crystal Growth on Locally Finite Partially Ordered Sets

TL;DR

This paper studies a Markovian growth process on partially ordered sets, providing bounds on passage times and a limit shape theorem, extending last passage percolation models to more general inhomogeneous settings.

Contribution

It introduces non-asymptotic bounds and a limit shape theorem for a generalized inhomogeneous exponential last passage percolation model on partially ordered sets.

Findings

Bounds on mean, variance, and moments of passage times

Limit shape theorem for monoid-structured sets

Methodology using backward equations and comparison inequalities

Abstract

We consider a Markovian growth process on a partially ordered set $\Lambda$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $\Lambda$. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice $\mathbb{N}_0^d$. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time $\tau_A$ to grow any set $A \subseteq \Lambda$ in terms of characteristics of $A$. We also give a limit shape theorem when $\Lambda$ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.