Existence of stationary ballistic deposition on the infinite lattice

TL;DR

This paper proves the existence of at least one invariant probability measure for a Markov process derived from a ballistic deposition model on an infinite lattice, contributing to understanding interface growth in the KPZ class.

Contribution

It establishes the existence of an invariant measure for the Markov process in the ballistic deposition model, a problem previously considered intractable.

Findings

Proved the existence of an invariant measure for the process.

Conjectured non-uniqueness of the invariant measure.

Provided partial evidence supporting the conjecture.

Abstract

Ballistic deposition is one of the many models of interface growth that are believed to be in the KPZ universality class, but have so far proved to be largely intractable mathematically. In this model, blocks of size one fall independently as Poisson processes at each site on the $d$-dimensional lattice, and either attach themselves to the column growing at that site, or to the side of an adjacent column, whichever comes first. It is not hard to see that if we subtract off the height of the column at the origin from the heights of the other columns, the resulting interface process is Markovian. The main result of this article is that this Markov process has at least one invariant probability measure. We conjecture that the invariant measure is not unique, and provide some partial evidence.