Renormalized Energy Concentration in Random Matrices
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Borodin_Renormalized energy.pdf
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Author(s)
Borodin, Alexei
Serfaty, Sylvia
Date Issued
April 2013
Journal
Communications in Mathematical Physics
Publisher
Springer-Verlag
Citation
Borodin, Alexei, and Sylvia Serfaty. “Renormalized Energy Concentration in Random Matrices.” Communications in Mathematical Physics 320, no. 1 (May 16, 2013): 199-244.
Version
Author's final manuscript
Abstract
We define a “renormalized energy” as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of Sandier and Serfaty (From the Ginzburg-Landau model to vortex lattice problems, 2012; 1D log-gases and the renormalized energy, 2013). Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix β-sine processes on the real line (β = 1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the β = 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.
Description
Author Manuscript October 23, 2012
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s00220-013-1716-z
