Local convergence in $t$-PNG

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 $\frac{1}{\hat{I}»(1-t)}$ and $\hat{I}»$, respectively, for some $\hat{I}»>0$. 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., Balázs and Seppäläinen (2009)), as well as blocking measure bounds like in Ferrari, Kipnis and Saada (1991).

21 pages, 6 figures; comments welcome