Loop equations for the semiclassical 2-matrix model with hard edges

TL;DR

This paper derives loop equations for the semiclassical 2-matrix model with hard edges, extending the framework to include rational potentials and arbitrary eigenvalue integration paths, including fixed endpoints.

Contribution

It formulates loop equations for the semiclassical 2-matrix model with hard edges, incorporating boundary contributions from fixed endpoints in the eigenvalue integration paths.

Findings

Loop equations are established for models with rational potentials.

Boundary contributions from hard edges are explicitly included.

The framework generalizes previous polynomial potential models.

Abstract

The 2-matrix models can be defined in a setting more general than polynomial potentials, namely, the semiclassical matrix model. In this case, the potentials are such that their derivatives are rational functions, and the integration paths for eigenvalues are arbitrary homology classes of paths for which the integral is convergent. This choice includes in particular the case where the integration path has fixed endpoints, called hard edges. The hard edges induce boundary contributions in the loop equations. The purpose of this article is to give the loop equations in that semicassical setting.