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Renormalized energy concentration in random matrices. (English) Zbl 1276.60007

The authors introduce a new definition of “renormalized energy” \(\mathcal{W}\) as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line, which is inspired by that of the renormalized energy \(W\) introduced in [E. Sandier and S. Serfaty, Commun. Math. Phys. 313, No. 3, 635–743 (2012; Zbl 1252.35034)] in the case of points in the plane and in [E. Sandier and S. Serfaty, “1D log gases and the renormalized energy: crystallization at vanishing temperature”, preprint, arXiv:1303.2968] in the case of points on the real line. The definitions for \(W\) and \(\mathcal{W}\) coincide when the point configuration has some periodicity. For the random matrix \(\beta\)-sine processes on the real line (\(\beta=1,2,4\)), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, the authors compute the expectation of the renormalized energy explicitly. Moreover, it is proved that for these processes the variance of the renormalized energy vanishes which shows concentration near the expected value.

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 1252.35034
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References:

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