Global Fluctuations of Random Matrices and the Second-Order Cauchy Transform
Diaz Torres, Mario
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In this thesis we study the global fluctuations of random matrices (i.e., the covariance of two traces) from a second-order free probability perspective, putting a particular emphasis on block Gaussian matrices. Our main contributions are threefold. First, we provide a formula for the second-order Cauchy transform of block Gaussian matrices in terms of the corresponding (matrix-valued) Cauchy transform. In order to do this, we introduce a (matrix-valued) second-order conditional expectation, in the spirit of the conditional expectations in operator-valued free probability theory. Second, we establish a criterion, based on the positivity of a cluster function, that guarantees the existence of a particular integral representation for the second-order Cauchy transform. This approach is based on some relations between the moments and fluctuation moments of a measure on the plane. Finally, we prove that under some conditions the covariance of resolvents converges to the second-order Cauchy transform. In fact, these conditions also ensure the convergence of the covariance of analytic linear statistics to a certain contour integral depending on the second-order Cauchy transform. In this context, we make explicit the relation between the notion of second-order limit distribution and the asymptotic Gaussianity of continuously differentiable linear statistics.