Global Fluctuations in General Beta Dyson Brownian Motion

Details Global Fluctuations in General Beta Dyson Brownian Motion Global Fluctuations in General Beta Dyson Brownian Motion

2006-10-25

arxiv.org/abs/math/0610750v1

Stochastic Processes and their Applications, Volume 118, Issue 6, June 2008, Pages 1022-1042

Mathematics

Probability

23 pages

Scientific paper

We consider a system of diffusing particles on the real line in a quadratic external potential and with repulsive electrostatic interaction. 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. 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.

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