Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0309058

Growth models, random matrices and Painleve transcendents

Peter J. Forrester
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter J. Forrester. 2003-09-29. Growth models, random matrices and Painleve transcendents. https://doi.org/10.1088/0951-7715%2F16%2F6%2F201

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local conservation laws, symmetries and Hamiltonian structure of three-dimensional incompressible Euler flow

All smooth local conservation laws, symmetries and adjoint symmetries of three-dimensional homogeneous incompressible Euler flow are classified in the primitive velocity--pressure variables. Explicit dependence on time and position and arbitrary finite orders of mixed time and spatial derivatives are allowed. Up to trivial currents, the conservation laws consist of incompressibility, linear and angular momentum, kinetic energy, helicity and generalised time-dependent momentum. Every local adjoint symmetry is represented by a multiplier of these currents. Every finite-order local symmetry is equivalent on solutions to a classical point symmetry, resolving a conjecture of Olver. An explicit local first-order Hamiltonian structure of the primitive system is also established in the geometric theory of partial differential equations. It maps the energy multiplier to time translation; its kernel on local adjoint symmetries is spanned by the helicity and incompressibility multipliers. The proof first excludes higher-order symmetries by comparing their responses to transverse and harmonic characteristic directions in compatible derivative data. The Hamiltonian map and its kernel then determine all local adjoint symmetries from the symmetry classification. Exact multiplier identities and the correspondence between multipliers and current classes complete the conservation-law classification.

math-ph↗
arXiv2026

Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies

We construct generalized Frobenius structures on Hurwitz spaces of arbitrary genus using extended primary differentials. Through contour integrals involving the Cauchy kernel, we derive explicit dispersionless Lax representations for the principal hierarchies of these structures and their almost-dual counterparts. We also construct solutions to the open WDVV equations for both the original and almost-dual structures, and derive explicit principal hierarchies for the resulting rank-one extensions. Finally, we show that these constructions admit Shramchenko deformations.

math-ph↗
arXiv2026

The dynamics through the first two Lagrange necks in the small-mass restricted three-body problem

We study the planar circular restricted three-body problem (PCR3BP) for sufficiently small mass ratios, from low energies to slightly above the second Lagrange value. We show that retrograde orbits around the primaries and Lyapunov orbits at the necks form the binding of a finite energy foliation. We construct these foliations using the theory of pseudoholomorphic curves in symplectizations developed by Hofer, Wysocki and Zehnder. Slightly above the second Lagrange value, the dynamics is transverse to families of disks bounded by the retrograde orbits, and cylinders connecting the retrograde orbits to the Lyapunov orbits, forming a $3-2-3-2$ foliation. The proof uses the rotating Kepler problem and Hill's lunar problem to control the singular limit as the mass ratio $μ\to 0$. As a consequence of the foliation, both Lyapunov orbits have infinitely many homoclinic orbits.

math-ph↗