Renormalized energy concentration in random matrices

Alexei Borodin (1) , Sylvia Serfaty (2)

1 MIT - Massachusetts Institute of Technology
2 LJLL - Laboratoire Jacques-Louis Lions

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 [SS1, SS3]. 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 con figuration 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.

Mots clés: Renormalized energy; Random point processes; Random matrices; Sine processes; Ginibre ensemble; Zeros of Gaussian analytic functions

Domaines: Analyse numérique [cs.NA]

Alexei Borodin, Sylvia Serfaty. Renormalized energy concentration in random matrices. 2012. ⟨hal-00667652⟩