Crystallization of Random Matrix Orbits

Vadim Gorin, Adam W. Marcus

Research output: Contribution to journalArticlepeer-review

Abstract

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

Original languageEnglish (US)
Pages (from-to)883-913
Number of pages31
JournalInternational Mathematics Research Notices
Volume2020
Issue number3
DOIs
StatePublished - Feb 7 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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