Local convergence in $t$-PNG

TL;DR

This paper proves local convergence of the $t$-PNG model with zero boundary conditions to its stationary version, confirming a recent conjecture, using coupling and microscopic concavity techniques.

Contribution

It establishes the local convergence of the $t$-PNG model with zero boundary to the stationary model, providing a rigorous proof for a conjecture from 2024.

Findings

Confirmed local convergence of $t$-PNG with zero boundary to stationary model

Developed coupling and microscopic concavity methods for proof

Extended techniques from particle models to $t$-PNG context

Abstract

We prove local convergence of the $t$-PNG model with zero boundary to the stationary $t$-PNG model, confirming a recent conjecture of Drillick and Lin (2024). The stationary $t$-PNG model is the one with both left and bottom boundaries of Poisson nucleations with rate parameters $\frac{1}{\lambda(1-t)}$ and $\lambda$, respectively, for some $\lambda>0$. In the proof, we consider the trajectories of certain second class particles via a basic monotone coupling of three $t$-PNG processes, and adapt microscopic concavity ideas used in particle models (e.g., Bal\'azs and Sepp\"al\"ainen (2009)), as well as blocking measure bounds like in Ferrari, Kipnis and Saada (1991).