Ballistic deposition patterns beneath a growing KPZ interface

TL;DR

This paper studies a (1+1)-dimensional ballistic deposition model within the KPZ universality class, introducing a variational formulation, identifying key structures, and defining a new 'hairy Airy process' with characterized scaling laws and critical exponents.

Contribution

It introduces a variational formulation for the ballistic deposition model, linking it to Burgers turbulence, and defines the novel 'hairy Airy process' with detailed statistical properties.

Findings

Identification of clusters and crevices with Burgers shocks and minimizers

Definition of the 'hairy Airy process' and its statistical properties

Derivation of exact critical exponents for scaling laws

Abstract

We consider a (1+1)-dimensional ballistic deposition process with next-nearest neighbor interaction, which belongs to the KPZ universality class, and introduce for this discrete model a variational formulation similar to that for the randomly forced continuous Burgers equation. This allows to identify the characteristic structures in the bulk of a growing aggregate ("clusters" and "crevices") with minimizers and shocks in the Burgers turbulence, and to introduce a new kind of equipped Airy process for ballistic growth. We dub it the "hairy Airy process" and investigate its statistics numerically. We also identify scaling laws that characterize the ballistic deposition patterns in the bulk: the law of "thinning" of the forest of clusters with increasing height, the law of transversal fluctuations of cluster boundaries, and the size distribution of clusters. The corresponding critical exponents are determined exactly based on the analogy with the Burgers turbulence and simple scaling considerations.