Crystal Growth on Locally Finite Partially Ordered Sets

Abstract

We consider a Markovian growth process on a partially ordered set $\hat{I}›$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $\hat{I}›$. 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 $\ddot{I}{„}_A$ to grow any set $A\subseteq \hat{I}›$ in terms of characteristics of $A$. We also give a limit shape theorem when $\hat{I}›$ 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.