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Free energy topological expansion for the 2-matrix model

Leonid Chekhov, Bertrand Eynard, Nicolas Orantin

TL;DR

This work provides a complete topological (large-N) expansion for the formal Hermitian two-matrix model by deriving a new product-form spectral curve and refined diagrammatic rules that depend only on branch-point data. It introduces the integration operators $H_x$ and $H_y$ to obtain the free energy from correlation functions, yielding a universal homogeneous relation for $\mathcal F^{(h)}$ across all moduli. The authors establish a residue-encoded representation of correlation functions and free energy via a cubic-vertex diagrammatic calculus, extending the one-matrix-model framework to the two-matrix setting and enabling closed-form expressions for all $h\ge 2$ (with $h=0,1$ known). The results deepen the link between algebraic geometry and matrix models, providing practical tools for computing higher-genus contributions and guiding future work on mixed correlators and broader potential classes.

Abstract

We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/ N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.

Free energy topological expansion for the 2-matrix model

TL;DR

This work provides a complete topological (large-N) expansion for the formal Hermitian two-matrix model by deriving a new product-form spectral curve and refined diagrammatic rules that depend only on branch-point data. It introduces the integration operators and to obtain the free energy from correlation functions, yielding a universal homogeneous relation for across all moduli. The authors establish a residue-encoded representation of correlation functions and free energy via a cubic-vertex diagrammatic calculus, extending the one-matrix-model framework to the two-matrix setting and enabling closed-form expressions for all (with known). The results deepen the link between algebraic geometry and matrix models, providing practical tools for computing higher-genus contributions and guiding future work on mixed correlators and broader potential classes.

Abstract

We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/ N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.

Paper Structure

This paper contains 22 sections, 6 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Definitions and algebraic-geometrical notation
  3. Definition of the model
  4. Notations
  5. Variable sets
  6. Resolvents
  7. The master loop equation
  8. Algebraic geometry notations
  9. Loop equations and fixed filling fractions
  10. A new formula for the spectral curve
  11. Diagrammatic rules for the correlation functions: new insight.
  12. Topological expansion of the free energy
  13. The $H$ operators
  14. Finding the free energy
  15. Dependence on other parameters
  16. ...and 7 more sections

Key Result

Theorem 3.1

The functions $E(x,y)$ and $U_0(p,y)$ can be written: and where the quotes $"\left<.\right>"$ mean that when we expand into cumulants, each time we find a 2-point function $\overline{w}_{2,0}$, we replace it by $w_{2,0}=\overline{w}_{2,0}+{1\over (x_1-x_2)^2}$ as in (defw2renorm).

Theorems & Definitions (14)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Theorem 4.1
  • Remark 5.1
  • Theorem 5.1
  • Lemma 5.1
  • Corollary 5.1
  • ...and 4 more