Open Access
2015 Empirical spacings of unfolded eigenvalues
Martin Venker, Kristina Schubert
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Electron. J. Probab. 20: 1-37 (2015). DOI: 10.1214/EJP.v20-4436


We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that the density of particles is asymptotically constant. This allows to consider all empirical spacings, where previous results were restricted to a tiny fraction of the particles. Moreover, we prove bounds on the rates of convergence. The main ingredient for the proof, a strong bulk universality result for correlation functions in the unfolded setting including optimal rates, should be of independent interest.


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Martin Venker. Kristina Schubert. "Empirical spacings of unfolded eigenvalues." Electron. J. Probab. 20 1 - 37, 2015.


Received: 20 July 2015; Accepted: 19 August 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1328.60014
MathSciNet: MR3425540
Digital Object Identifier: 10.1214/EJP.v20-4436

Primary: 60B20
Secondary: 42C05 , 82C22

Keywords: Empirical Spacings , Gaudin Distribution , random matrices , rates of convergence , Repulsive Particle Systems , Unfolding

Vol.20 • 2015
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