Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

By M. Katori, H. Tanemura. 2004

152 citations recorded by the source corpus.

As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0e735e44fdf3055a4ae235b80578a041eeb6208c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. W. Koenig
2. Peter Eichelsbacher
3. J. Małecki
4. Piotr Graczyk
5. Wolfgang Konig
6. Sourav Chatterjee

In Discrete Mathematics and Combinatorics

1. Peter Eichelsbacher
2. Wolfgang Konig
3. W. Koenig
4. J. Małecki
5. Piotr Graczyk
6. Vitali Wachtel
7. Denis Denisov

In Statistics and Probability

1. Peter Eichelsbacher
2. Wolfgang Konig
3. W. Koenig
4. J. Małecki
5. Piotr Graczyk
6. Vitali Wachtel
7. Denis Denisov
8. Sourav Chatterjee

Citations in the corpus

• Orthogonal polynomial ensembles in probability theory (2004, background): “(iv) Further important examples with physical relevance are derived in [ KT04 ]; in fact, process versions of all ten classes of Gaussian random matrices mentioned at the beginning of Section 2 are analysed, and their eigenvalue processes are characterised in terms of non-colliding diusions.”
• Multidimensional Yamada-Watanabe theorem and its applications to particle systems (2011, methodology): “We apply Theorems 1, 5 and 6 to (i) noncolliding particle systems of squared Bessel processes which are intensely studied in recent years in statistical and mathematical physics (Katori, Tanemura [14, 15, 16, 17]).”
• Ordered Random Walks (2006, methodology): “Non-colliding Brownian motions on the circle were investigated in [HW96], asymptotic questions about Brownian motions in a Weyl chamber in [Gr99], and a systematic study of a large class of physically relevant matrix-valued processes and their eigenvalue processes is carried out in [ KT04 ].”