# Schwinger-Dyson and loop equations for a product of square Ginibre random matrices

## TL;DR

This paper derives loop equations for the resolvents of a product of two Ginibre matrices using Schwinger-Dyson techniques, providing insights into their spectral properties and corrections beyond the large N limit.

## Contribution

It introduces a novel approach to derive loop equations for matrix products without reformulating in terms of singular values, and analyzes the spectral curve geometry.

## Key findings

- Large N limit of the 2-point correlation function's Stieltjes transform
- First correction to the density's Stieltjes transform
- Explicit results for various resolvents and their corrections

## Abstract

In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large $N$ limit of the Stieltjes transform of the $2$-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.