Correlations for the circular Dyson brownian motion model with Poisson initial conditions

doi:10.1016/S0550-3213(98)00551-3

Nuclear Physics B

Volume 532, Issue 3, 9 November 1998, Pages 733-752

https://doi.org/10.1016/S0550-3213(98)00551-3 Get rights and content

Abstract

The circular Dyson brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the initial condition that the particles are non-interacting (Poisson statistics). Jack polynomial theory is used to derive a simple exact expression for the density-density correlation with the position of one particle specified in the initial state, and the position of one particle specified at time τ, valid for all β > 0. The same correlation with two particles specified in the initial state is also derived exactly, and some special cases of the theoretical correlations are illustrated by comparison with the empirical correlations calculated from the eigenvalues of certain parameter-dependent Gaussian random matrices. Application to fluctuation formulas for time-displaced linear statistics in made.

References (23)

P.J. Forrester
Physica A
(1996)
C.W.J. Beenakker et al.
Physica A
(1994)
E. Brézin et al.
Nucl. Phys. B
(1996)
A. Pandey
Chaos, Solitons & Fractals
(1995)
P.J. Forrester
Mod. Phys. Lett. B
(1995)
P.J. Forrester et al.
Physica A
(1997)
P.J. Forrester et al.
Nucl. Phys. B
(1996)
H. Spohn
F.J. Dyson
J. Math. Phys.
(1962)
P.J. Forrester et al.
J. Stat. Phys.
(1997)

F. Haake

Quantum Signatures of Chaos

(1992)

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Correlations for the circular Dyson brownian motion model with Poisson initial conditions☆

Abstract

Physica A

Physica A

Nucl. Phys. B

Chaos, Solitons & Fractals

Mod. Phys. Lett. B

Physica A

Nucl. Phys. B

J. Math. Phys.

J. Stat. Phys.

Quantum Signatures of Chaos