Growth models, random matrices and Painleve transcendents

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 Prähofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlevé II transcendent plays a prominent role.

27 pages, 5 figures

References in corpus (4)

• Exact scaling functions for one-dimensional stationary KPZ growth
• Simple strong glass forming models: mean-field solution with activation
• Discrete Painlevé equations and random matrix averages
• Interpretations of some parameter dependent generalizations of classical matrix ensembles

Cited by in corpus (7)

• Universality for mathematical and physical systems
• Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• Symmetrized models of last passage percolation and non-intersecting lattice paths
• Electrostatics of Coulomb gas, lattice paths, and discrete polynuclear growth
• Level-spacing distributions of the Gaussian unitary random matrix ensemble
• Asymptotic Behaviours Given by Elliptic Functions in $P_I$--$P_V$