arXiv subjects
Publications and source records attributed to Florent Bekerman.
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.
We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $\beta$-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.
We prove that the linear statistics of eigenvalues of $β$-log gasses satisfying the one-cut and off-critical assumption with a potential $V \in C^6(\mathbb{R})$ satisfy a central limit theorem at all mesoscopic scales $α\in (0; 1)$. We prove this for compactly supported test functions $f \in C^5(\mathbb{R})$ using loop equations at all orders along with rigidity estimates.
We construct approximate transport maps for non-critical Beta-matrix models, that is, maps so that the push forward of a non-critical Beta-matrix model with a given potential is a non-critical Beta-matrix model with another potential, up to a small error in the total variation distance. As a consequence, we deduce that local statistics have the same asymptotic behavior for both models.