On the global fluctuations of block Gaussian matrices

Abstract

In this paper we study the global fluctuations of block Gaussian matrices within the framework of second-order free probability theory. In order to compute the second-order Cauchy transform of these matrices, we introduce a matricial second-order conditional expectation and compute the matricial second-order Cauchy transform of a certain type of non-commutative random variables. As a by-product, using the linearization technique, we obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices.

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

“…We are therefore tempted to speculate that, by combining second order freeness and freeness with amalgamation [74], the notion of the non-orthogonality of eigenvectors can be extended into a broader context of operators in von Neumann algebras. Indeed, an equation similar to (47) has recently appeared in the description of fluctuations of Gaussian block matrices [75]. Moreover, the diagrammatic calculations of the traced product of resolvents resemble the partition structure diagrams introduced in [76].…”

On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fréchet Variation

“…Furthermore, we establish that the covariance of resolvents converges to this transform (see Corollary 8) and that the limiting covariance of analytic linear statistics can be expressed as a contour integral depending on the second order Cauchy transform, see Theorem 11. Since the second-order Cauchy transform of block Gaussian matrices was recently found in [14], these results provide an effective way to compute the covariance of analytic linear statistics of block Gaussian matrices.…”

“…In [14], Diaz et al recently found a formula for the second-order Cauchy transform of block Gaussian matrices. Specifically, they derive a formula at the level of formal expressions and then extend it to the analytic level.…”

“…This shows that block Gaussian matrices satisfy A2. As mentioned in the introduction, the second-order Cauchy transform of block Gaussian matrices was recently found in [14], see equation (5).…”

On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices

“…Under this framework, CLTs for linear eigenvalue statistics of Wigner matrices with general variance profiles were obtained in [2]. Global fluctuations of block Gaussian matrices with variance profiles were proved within the framework of second-order free probability theory, see [25] and references therein. In addition, CLTs on global scales for large sample covariance matrices given a general variance profile were discussed in [45].…”