Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles

Abstract

Dyson's model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant $\beta /2$. We give sufficient conditions for initial configurations so that Dyson's model with $\beta =2$ and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of $\mathbb{Z}$ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

v3: AMS-LaTeX, 32 pages, 1 figure, revised version for publication in Commun. Math. Phys

References in corpus (3)

• Differential Equations for Dyson Processes
• Noncolliding Brownian Motion and Determinantal Processes
• Polynuclear growth model, GOE${}^2$ and random matrix with deterministic source

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