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Global fluctuations in general [beta] Dyson's Brownian motion

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  • Bender, Martin

Abstract

We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.

Suggested Citation

  • Bender, Martin, 2008. "Global fluctuations in general [beta] Dyson's Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1022-1042, June.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:6:p:1022-1042
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    References listed on IDEAS

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    1. Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
    2. Israelsson, Stefan, 2001. "Asymptotic fluctuations of a particle system with singular interaction," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 25-56, May.
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