One-matrix differential reformulation of two-matrix models

TL;DR

This paper introduces a differential reformulation of two-matrix models that enables diagonalization and leads to a new determinant expression, potentially advancing computational methods in matrix models.

Contribution

It develops a one-matrix differential formulation for two-matrix models and derives a novel determinant representation of their partition function.

Findings

Derived a differential reformulation for two-matrix models.

Established equivalence between differential and integral formulations.

Proposed potential applications to orthogonal polynomial methods.

Abstract

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.