Hydrodynamics of the $t$-PNG model via a colored $t$-PNG model

TL;DR

This paper establishes the hydrodynamic limit of the deformed $t$-PNG model using a colored version and ergodic theorems, extending previous results and introducing new techniques for analyzing growth models.

Contribution

It introduces a colored $t$-PNG model to prove the hydrodynamic limit and derives the limiting constant through a law of large numbers for $eta$-points.

Findings

Hydrodynamic limit of the $t$-PNG model is proven.

A colored $t$-PNG model is constructed for analysis.

A law of large numbers for $eta$-points is established.

Abstract

The $t$-PNG model introduced in Aggarwal, Borodin, and Wheeler (2021) is a deformed version of the polynuclear growth (PNG) model. In this paper, we prove the hydrodynamic limit of the model using soft techniques. One key element of the proof is the construction of a colored version of the $t$-PNG model, which allows us to apply the superadditive ergodic theorem and obtain the hydrodynamic limit, albeit without identifying the limiting constant. We then find this constant by proving a law of large numbers for the $\alpha$-points, which generalizes Groeneboom (2001). Along the way, we construct the stationary $t$-PNG model and prove a version of Burke's theorem for it.