A Theory of Singular Values for Finite Free Probability

Aurelien Gribinski ()
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Aurelien Gribinski: Princeton University

Journal of Theoretical Probability, 2024, vol. 37, issue 2, 1257-1298

Abstract: Abstract We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. This study is motivated by the companion papers Gribinski (J Comb Theory. arXiv:1904.11552 ; Existence and polynomial time construction of biregular, bipartite Ramanujan graphs of all degrees. arXiv:2108.02534 ) , as well as the corresponding paper dealing with the square case (Marcus in Polynomial convolutions and (finite) free probability, 2021. arXiv preprint arXiv:2108.07054 ). In the process we exhibit a canonic bivariate operation on polynomials, seemingly more natural when singular values are concerned. We show that we can replicate the transforms from free probability and that asymptotically there is convergence from rectangular finite free probability to rectangular free probability. Lastly, we show that classical distribution results such as a law of large numbers or a central limit theorem can be made explicit in this new framework where random variables are replaced by polynomials.

Date: 2024
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DOI: 10.1007/s10959-023-01295-0

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