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May 2018 Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
Kurt Johansson, Gaultier Lambert
Ann. Probab. 46(3): 1201-1278 (May 2018). DOI: 10.1214/17-AOP1178


We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the correlation kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we obtain a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we observe a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.


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Kurt Johansson. Gaultier Lambert. "Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes." Ann. Probab. 46 (3) 1201 - 1278, May 2018.


Received: 1 July 2015; Revised: 1 October 2016; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894774
MathSciNet: MR3785588
Digital Object Identifier: 10.1214/17-AOP1178

Primary: 60B20, 60F05, 60G55

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • May 2018
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