## Renormalized Energy Concentration in Random Matrices

##### Author(s)

Borodin, Alexei; Serfaty, Sylvia
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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

##### Date issued

2013-04##### Department

Massachusetts Institute of Technology. Department of Mathematics##### 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

##### ISSN

0010-3616

1432-0916