Crystallization of random matrix orbits
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1706.07393.pdf
Description
Accepted version
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342.79 KB
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Adobe PDF
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491a7e92315e5bcf69d97ec3d1a357ca
Author(s)
Gorin, Vadim
Marcus, Adam W.
Date Issued
February 2020
Journal
International mathematics research notices
Publisher
Oxford University Press (OUP)
Citation
Gorin, Vadim, and Adam W. Marcus, "Crystallization of random matrix orbits." International mathematics research notices 2020, 3 (February 2020): p. 883-913 doi 10.1093/imrn/rny052 ©2020 Author(s)
Version
Author's final manuscript
Abstract
Three operations on eigenvalues of real/complex/quaternion (corresponding to β=1,2,4) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of β>0 through associated special functions. We show that the β→∞ 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 β self-adjoint matrix with fixed eigenvalues is known as the β-corners process. We show that as β→∞ 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. ©2020
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNY052
