Deformed Polynuclear Growth in $(1+1)$ Dimensions

TL;DR

This paper introduces a deformed polynuclear growth model called the $t$-PNG, which interpolates between the standard PNG and a model with errors in sorting, and studies its asymptotic behavior and phase transitions.

Contribution

It defines the $t$-PNG model, proves Tracy-Widom GUE asymptotics for fixed $t$, and shows convergence to KPZ solutions as $t$ approaches 1, using solvable stochastic models.

Findings

$t$-PNG exhibits Tracy-Widom GUE asymptotics for $t<1$.

As $t$ approaches 1, the model converges to the KPZ equation solution.

External source distributions can induce phase transitions similar to Baik-Ben Arous-Peche.

Abstract

We introduce and study a one parameter deformation of the polynuclear growth (PNG) in $(1+1)$-dimensions, which we call the $t$-PNG model. It is defined by requiring that, when two expanding islands merge, with probability $t$ they sprout another island on top of the merging location. At $t=0$, this becomes the standard (non-deformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the $t$-PNG model allows errors to occur in the sorting algorithm with probability $t$. We prove that the $t$-PNG model exhibits one-point Tracy-Widom GUE asymptotics at large times for any fixed $t\in [0,1)$, and one-point convergence to the narrow wedge solution of the Kardar-Parisi-Zhang (KPZ) equation as $t$ tends to $1$. We further construct distributions for an external source that are likely to induce Baik-Ben Arous-Peche type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions.