Polynuclear growth model with external source and random matrix model with deterministic source

Abstract

We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE2 , which is defined as the square of the Gaussian orthogonal ensemble (GOE) Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE2 as "the square of GOE," ours has an advantage in that it can also describe the transition from the Gaussian unitary ensemble (GUE) Tracy-Widom distribution to the GOE2 . We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding Brownian motion model and the multilayer PNG model with an external source. This provides us with a multimatrix model interpretation of the multipoint height distributions of the PNG model with an external source.