Growth models, random matrices and Painleve transcendents

TL;DR

This paper explores the Hammersley process and its variants, connecting growth models, random matrices, and Painleve transcendents, and reviews their mathematical derivations and universality properties.

Contribution

It reviews the derivation of formulas for the Hammersley process with symmetries and boundary sources, highlighting the role of Painleve II transcendents in the KPZ universality class.

Findings

Distribution formulas involve unitary group averages

Symmetric variants allow similar distribution formulas

Painleve II transcendent characterizes the scaled distribution

Abstract

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlev\'e II transcendent plays a prominent role.