Fluctuations of a one-dimensional polynuclear growth model in a half space

TL;DR

This paper analyzes the height fluctuations in a one-dimensional polynuclear growth model confined to a half space, revealing connections to random matrix theory and eigenvalue distributions at the soft edge.

Contribution

It establishes a link between the growth model's fluctuations and the eigenvalue statistics of orthogonal/symplectic ensembles in random matrix theory, providing new asymptotic results.

Findings

Fluctuations near the origin match the largest eigenvalue distributions in random matrix ensembles.

The multi-layer model reduces to a determinantal process for specific nucleation rates.

Asymptotic analysis reveals universal behavior akin to random matrix theory at the soft edge.

Abstract

We consider the multi-point equal time height fluctuations of a one-dimensional polynuclear growth model in a half space. For special values of the nucleation rate at the origin, the multi-layer version of the model is reduced to a determinantal process, for which the asymptotics can be analyzed. In the scaling limit, the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory.