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The Wigner semi-circle law and eigenvalues of matrix-valued diffusions
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  • Published: June 1992

The Wigner semi-circle law and eigenvalues of matrix-valued diffusions

  • Terence Chan1 nAff2 

Probability Theory and Related Fields volume 93, pages 249–272 (1992)Cite this article

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Summary

A system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent Ornstein-Uhlenbeck processes is derived. This corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter-particle repulsion. The associated empirical distribution of particles is shown to converge weakly (as the number of particles tends to infinity) to a limiting measure-valued process which may be characterized as the weak solution of a deterministic ODE. The Wigner semi-circle density is found to be one of the equilibrium points of this limiting equation.

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Author information

Author notes
  1. Terence Chan

    Present address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, EH14 4AS, Edinburgh, UK

Authors and Affiliations

  1. Statistical Laboratory, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, UK

    Terence Chan

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  1. Terence Chan
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Cite this article

Chan, T. The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probab. Th. Rel. Fields 93, 249–272 (1992). https://doi.org/10.1007/BF01195231

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  • Received: 15 January 1990

  • Revised: 28 December 1991

  • Issue Date: June 1992

  • DOI: https://doi.org/10.1007/BF01195231

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Keywords

  • Stochastic Process
  • Weak Solution
  • Equilibrium Point
  • Diffusion Model
  • Mathematical Biology
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