A Functional Limit Theorem for the Sine-Process

Abstract

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 sine-process for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity 1/2. Contents

A method for solving the limit of multivariate composite functions

“…It is shown that the time integral of the sinusoidal process can be approximated as a sum of a linear Gaussian process and independent Gaussian fluctuations. It is proved that the two types of linear statistics are variance-growing and variance-bounded [12]. Scholars such as Magdziarz, M and others propose an alternative definition of ctrw with continuous trajectories and give a scalar limit theorem for this random walk sequence.…”