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PDE's for the Dyson, Airy and Sine processes

Mark Adler — 2005

Annales de l’institut Fourier

In 1962, Dyson showed that the spectrum of a $n \times n$ random Hermitian matrix, whose entries (real and imaginary) diffuse according to $n^{2}$ independent Ornstein-Uhlenbeck processes, evolves as $n$ non-colliding Brownian particles held together by a drift term. When $n \to \infty$ , the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...

The Intersection of Four Quadrics in IP6, Abelian Surfaces and their Moduli.

Mark Adler; Pierre van Moerbeke

Mathematische Annalen

The complex geometry of the Kowalewski-Painlevé analysis.

Mark Adler; Pierre van Moerbeke — 1989

Inventiones mathematicae

The Dyson Brownian Minor Process

Mark Adler; Eric Nordenstam; Pierre Van Moerbeke — 2014

Annales de l’institut Fourier

Consider an $n \times n$ Hermitean matrix valued stochastic process ${H_{t}}_{t \geq 0}$ where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k \times k$ minors in the upper left corner of $H_{t}$ . Projecting...

The Toda lattice, Dynkin diagrams, singularities and Abelian varieties.

Mark, Moerbeke, Pierre van Adler — 1991

Inventiones mathematicae

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