Transport maps for β-matrix models in the multi-cut regime

Abstract

We use the transport methods developped in [3] to obtain universality results for local statistics of eigenvalues in the bulk and at the edge for β-matrix models in the multi-cut regime. We construct an approximate transport map inbetween two probability measures from the fixed filling fraction model discussed in [6] and deduce from it universality in the initial model.

Mesoscopic central limit theorem for general $\beta$-ensembles

“…When we integrate the term on the third line of (2.30) against M and both terms appearing in M (2) N above are of the same form as g so they are bounded by CN ξ . Returning to (2.31), we may bound…”

“…Recently, great progress has been made to understand the behaviour of β-log gasses. At the microscopic scale, the eigenvalues exhibit a universal behaviour (see [3], [7], [6], [2]) and the local statics of the eigenvalues are described by the Sine β process in the bulk and the Stochastic Airy Operator at the edge (see [19] and [17] for definitions). At the macroscopic level, the eigenvalues satisfy a central limit theorem and the re-centered linear statistics of the eigenvalues converge towards a Gaussian random variable.…”

Free analysis and random matrices

“…Such a result was also proved in [11,12] under weaker assumptions on V (but forˇ 1) and by Scherbina [41] (in the bulk but including cases where the support of V is not connected, an extension that was also considered in [3] by using approximate transport maps).…”

“…But in fact Schwinger-Dyson equation can be used even further to find a complete expansion for EOETr.P / in terms of N 1 . Indeed, the large dimension behavior of such systems can be analyzed thanks to (3), by expanding the observables around their limit, in the spirit of Stein method or, more simply, perturbation theory. These are called topological expansion as in the case of random matrices, each coefficient of the series corresponds to a generating function for the enumeration of maps of a given genus.…”

A transportation approach to universality in random matrix theory

“…Then, in the cases β = 1, 4, universality was proved in [10,11,22,28,29] (see also [12] for a review). Finally, the local fluctuations of more general β-ensembles were only derived recently [27,37] in the Gaussian case, and universality for general β-ensembles was obtained in [3][4][5][6]17,20,21,30].…”