## The Annals of Probability

### Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 46, Number 3 (2018), 1201-1278.

Dates
Revised: October 2016
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.aop/1523520017

Digital Object Identifier
doi:10.1214/17-AOP1178

Mathematical Reviews number (MathSciNet)
MR3785588

Zentralblatt MATH identifier
06894774

#### Citation

Johansson, Kurt; Lambert, Gaultier. Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes. Ann. Probab. 46 (2018), no. 3, 1201--1278. doi:10.1214/17-AOP1178. https://projecteuclid.org/euclid.aop/1523520017

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