Fluctuation formula for complex random matrices
P.J. Forrester (University of Melbourne)
TL;DR
This paper proves a Gaussian fluctuation formula for linear statistics of complex random matrices, highlighting the effects of symmetry and boundary contributions on the distribution's variance.
Contribution
It establishes a Gaussian fluctuation formula for rotationally invariant linear statistics and extends predictions to non-symmetric cases using Coulomb gas theory.
Findings
Gaussian fluctuation formula for rotationally invariant statistics
Variance decomposes into bulk and boundary contributions
Boundary effects influence long-range correlations
Abstract
A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict that the distribution will again be a Gaussian, with a specific mean and variance. The variance splits naturally into a bulk and surface contibution, the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Random Matrices and Applications