Tatyana Shcherbyna

University of Wisconsin - Madison

Supersymmetric approach to the deformed Ginibre ensemble

Mathematical Physics Seminar

25th March 2022, 2:30 pm – 3:30 pm
Fry Building, 4th Floor Seminar Room

Supersymmetric approach to random matrix theory is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density of eigenvalues, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult. In this talk we will discuss some recent progress in application of SUSY to the deformed Ginibre ensemble. Namely, we consider non-Hermitian random $n\times n$ matrices of the form $H=A+H_0$, where $H_0$ is a standard Ginibre matrix with iid Gaussian entries, and $A$ is a rather general matrix (deterministic or random). Such matrices are important in communication theory, where $A$ is considered as a ``signal", and $H_0$ as a ``noise" matrix. In particular, one is interested in effective numerical solvability of a large system of linear equations $(H-z) x=b$ which is determined by the behavior of the smallest singular value $\sigma_1(H-z)$ of $H-z$. The classical bound of Sankar, Spielman and Teng states that the smallest singular value $\sigma_1(H-z)$ is of order not smaller than $n^{-1}$. Using SUSY, we show that if $z$ is around the spectral edge of $H$, then this bound can be improved. The talk is based on a joint work with Mariya Shcherbina.


Talk recording: https://mediasite.bris.ac.uk/Mediasite/Play/783361b9f40c4f109edfc0272582377d1d

Organiser: Thomas Bothner

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