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Complex Brownian motion representation of the Dyson model. (English) Zbl 1306.60119

Summary: Dyson’s Brownian motion model with the parameter \(\beta=2\), which we simply call the Dyson model in the present paper, is realized as an \(h\)-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on the initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of the \(h\)-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.

MSC:

60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G44 Martingales with continuous parameter
82C22 Interacting particle systems in time-dependent statistical mechanics
32A15 Entire functions of several complex variables