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

TL;DR

This paper applies transport methods to establish universality of eigenvalue statistics in multi-cut beta-matrix models, both in the bulk and at the edge, by constructing an approximate transport map between measures.

Contribution

It introduces a novel application of transport maps to prove universality in multi-cut beta-matrix models, extending previous methods to more complex regimes.

Findings

Universality of local eigenvalue statistics in multi-cut regimes

Construction of an approximate transport map between probability measures

Extension of universality results to bulk and edge eigenvalues

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 $\beta$-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.