Elliptic Determinantal Processes and Elliptic Dyson Models

Abstract

We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families $A_{N-1}$, $B_N$, $C_N$ and $D_N$, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.

References in corpus (7)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Nonintersecting Brownian excursions
• Zeros of Airy Function and Relaxation Process
• Weight-dependent commutation relations and combinatorial identities
• Derivation of a $B{C}_n$ elliptic summation formula via the fundamental invariants
• A generalization of the Sears--Slater transformation and elliptic Lagrange interpolation of type $B{C}_n$
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Cited by in corpus (4)

• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes
• Polyanalytic Reproducing Kernels on the Quantized Annulus
• Excursion Processes Associated with Elliptic Combinatorics