Fluctuations of the one-dimensional polynuclear growth model with external sources

TL;DR

This paper analyzes the multi-point height fluctuations in a one-dimensional polynuclear growth model with external sources, revealing new transition kernels between known distributions and generalizing previous results.

Contribution

It introduces explicit kernels describing transitions between classical fluctuation distributions in the model, extending prior one-point distribution results to multi-point correlations.

Findings

Derived explicit Fredholm kernels for multi-point fluctuations.

Identified new transition kernels between GOE$^2$, GUE Tracy-Widom, and Gaussian distributions.

Confirmed that multi-point fluctuation results match previous Riemann-Hilbert analysis.

Abstract

The one-dimensional polynuclear growth model with external sources at edges is studied. The height fluctuation at the origin is known to be given by either the Gaussian, the GUE Tracy-Widom distribution, or certain distributions called GOE$^2$ and $F_0$, depending on the strength of the sources. We generalize these results and show that the scaling limit of the multi-point equal time height fluctuations of the model are described by the Fredholm determinant, of which the limiting kernel is explicitly obtained. In particular we obtain two new kernels, describing transitions between the above one-point distributions. One expresses the transition from the GOE$^2$ to the GUE Tracy-Widom distribution or to the Gaussian; the other the transition from $F_0$ to the Gaussian. The results specialized to the fluctuation at the origin are shown to be equivalent to the previously obtained ones via the Riemann-Hilbert method.