Elliptic Determinantal Process of Type A

Abstract

We introduce an elliptic extension of Dyson's Brownian motion model, which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle. Using elliptic determinant evaluations related to the reduced affine root system of types $A$, we give determinantal martingale representation (DMR) for the process, when it is started at the configuration with equidistant spacing on the circle. DMR proves that the process is determinantal and the spatio-temporal correlation kernel is determined. By taking temporally homogeneous limits of the present elliptic determinantal process, trigonometric and hyperbolic versions of noncolliding diffusion processes are studied.

v5: AMS-LaTeX, 35 pages, no figure, references updated for publication in Probab. Theory Relat. Fields

References in corpus (5)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Littelmann paths and brownian paths
• Determinantal Martingales and Noncolliding Diffusion Processes
• Nonintersecting lattice paths on the cylinder
• Elliptically Distributed Lozenge Tilings of a Hexagon

Cited by in corpus (8)

• Two limiting regimes of interacting Bessel processes
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes
• Elliptic Determinantal Processes and Elliptic Dyson Models
• Exact solution of interacting particle systems related to random matrices
• Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes
• Determinantal Martingales and Correlations of Noncolliding Random Walks
• Excursion Processes Associated with Elliptic Combinatorics