Transport Maps for $β$-Matrix Models in the Multi-Cut Regime

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

“…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.…”

“…2 3 +ξ , by our choice of ξ. When we substitute γ i by x, the same argument holds becauseN α |x − γ i | ≤ N α |γ i − γ i+1 |, which is of order N α−1 on J 2 (as we showed in statement (a)) and of order CN α− 2 3 on J 3 .…”

Universality in several-matrix models via approximate transport maps

“…Universality in the β-ensembles was first addressed in [BEY14a] (in the bulk, β > 0, V ∈ C 4 ), then in [BEY14b] (at the edge, β ≥ 1, V ∈ C 4 ), [KRV13] (at the edge, β > 0, V convex polynomial), and finally in [Shc14a] (in the bulk, β > 0, V analytic, multi-cut case included) and in [BFG15] (in the bulk and the edge, V smooth enough). The universality at the edge in the several-cut case is treated in [Bek15]. The case where the interaction is more general than a Coulomb gas, but given by a mean-field interaction i<j ϕ(x i − x j ) where ϕ(t) behaves as |t| β in a neighborhood of the origin and log |x| −β ϕ(x) is real analytic as well as the potential, was considered in [GV14] (β = 2, universality in the bulk), [Ven13] (β > 0, universality in the bulk), and [KV15] (β = 2, universality at the edge).…”