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A functional limit theorem for the sine-process

International Mathematics Research Notices. 2019. Vol. 2019. No. 1. P. 249-319.
Bufetov A. I., Dymov A. V.

The main result of this paper is a functional limit theorem for the sine-process. In
particular, we study the limit distribution, in the space of trajectories, for the number
of particles in a growing interval. The sine-process has the Kolmogorov property and
satisfies the central limit theorem, but our functional limit theorem is very different
from the Donsker Invariance Principle. We show that the time integral of our process
can be approximated by the sum of a linear Gaussian process and independent Gaussian
fluctuations whose covariance matrix is computed explicitly. We interpret these results
in terms of the Gaussian free field convergence for the random matrix models. The proof
relies on a general form of the multidimensional central limit theorem under the sineprocess for linear statistics of two types: those having growing variance and those with
bounded variance corresponding to observables of Sobolev regularity 1/2.