Asymptotic analysis of random matrices with external source and a family of algebraic curves

TL;DR

This paper develops a framework for analyzing the asymptotic behavior of random matrices with two eigenvalues in the source term, linking it to algebraic curves, and applies it to quartic external fields to derive detailed eigenvalue distributions.

Contribution

It establishes conditions equivalent to the existence of a specific algebraic curve for asymptotic analysis, and proves such a curve exists for quartic external fields, enabling precise eigenvalue distribution results.

Findings

Derived asymptotic eigenvalue density for quartic external fields

Proved existence of algebraic curve for the model

Established bulk and edge universality results

Abstract

We present a set of conditions which, if satisfied, provide for a complete asymptotic analysis of random matrices with source term containing two distinct eigenvalues. These conditions are shown to be equivalent to the existence of a particular algebraic curve. For the case of a quartic external field, the curve in question is proven to exist, yielding precise asymptotic information about the limiting mean density of eigenvalues, as well as bulk and edge universality.