Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

TL;DR

This paper analyzes the long-term behavior of the PNG and parallel TASEP models in 1+1 dimensions, showing convergence to the Airy_1 process along space-like paths, with implications for understanding surface growth phenomena.

Contribution

It establishes the convergence of surface height distributions to the Airy_1 process for PNG and TASEP models along space-like paths, extending previous results to more general observation points.

Findings

Surface height distributions converge to the Airy_1 process.

Results hold for any space-like path in space-time.

Provides explicit joint distribution formulas for the models.

Abstract

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy_1 process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.