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Notes on Matrix Models

Dionysios Anninos, Beatrix Mühlmann

TL;DR

This work surveys the landscape of large-$N$ matrix models, starting from vector integrals and saddle-point methods, through one- and two-matrix models, to quantum-mechanical matrix theories and their continuum limits. It develops a coherent toolkit— saddle-point analysis, the resolvent, loop equations, and orthogonal-polynomial techniques— to access planar and non-planar (genus) contributions, including non-perturbative effects via double-scaling limits and Painlevé equations. A central theme is the connection between discrete matrix-model perturbations and continuum 2D gravity, Liouville theory, and KPZ scaling, with detailed discussions of how criticality, multicriticality, and instanton sectors encode continuum physics. The notes further extend to supersymmetric variants, 2D Yang–Mills, Chern–Simons/topological strings, and modern non-matrix large-$N$ theories such as SYK/tensor models, highlighting potential holographic interpretations and future research directions in emergent spacetime and quantum gravity.

Abstract

In these notes we explore a variety of models comprising a large number of constituents. An emphasis is placed on integrals over large Hermitian matrices, as well as quantum mechanical models whose degrees of freedom are organised in a matrix-like fashion. We discuss the relation of matrix models to two-dimensional quantum gravity coupled to conformal matter. We provide a brief overview of a variety of more general quantum mechanical matrix models and their putative worldsheet interpretation, as well as other recent developments on large $N$ systems. We end with a discussion of open questions and future directions.

Notes on Matrix Models

TL;DR

This work surveys the landscape of large- matrix models, starting from vector integrals and saddle-point methods, through one- and two-matrix models, to quantum-mechanical matrix theories and their continuum limits. It develops a coherent toolkit— saddle-point analysis, the resolvent, loop equations, and orthogonal-polynomial techniques— to access planar and non-planar (genus) contributions, including non-perturbative effects via double-scaling limits and Painlevé equations. A central theme is the connection between discrete matrix-model perturbations and continuum 2D gravity, Liouville theory, and KPZ scaling, with detailed discussions of how criticality, multicriticality, and instanton sectors encode continuum physics. The notes further extend to supersymmetric variants, 2D Yang–Mills, Chern–Simons/topological strings, and modern non-matrix large- theories such as SYK/tensor models, highlighting potential holographic interpretations and future research directions in emergent spacetime and quantum gravity.

Abstract

In these notes we explore a variety of models comprising a large number of constituents. An emphasis is placed on integrals over large Hermitian matrices, as well as quantum mechanical models whose degrees of freedom are organised in a matrix-like fashion. We discuss the relation of matrix models to two-dimensional quantum gravity coupled to conformal matter. We provide a brief overview of a variety of more general quantum mechanical matrix models and their putative worldsheet interpretation, as well as other recent developments on large systems. We end with a discussion of open questions and future directions.

Paper Structure

This paper contains 50 sections, 382 equations, 19 figures, 3 tables.

Table of Contents

  1. Introduction$^{\color{magenta} A, B}$
  2. Large $N$ vector integrals
  3. Saddle point approximation
  4. Vector integrals
  5. A perturbative expansion: the Cactus diagrams
  6. Diagrams resummed
  7. Large $N$ integrals over a single matrix I$^{\color{magenta} C}$
  8. Eigenvalue distribution
  9. Saddle point approximation $\&$ the resolvent
  10. General polynomial $\&$ multicritical models
  11. Large $N$ factorisation $\&$ loop equations
  12. A perturbative expansion: the Riemann surfaces
  13. Large $N$ integrals over a single matrix II$^{\color{magenta}C,D}$
  14. Orthogonal polynomials
  15. Non-planar contributions
  16. ...and 35 more sections

Figures (19)

  • Figure 1: Propagator and quartic vertex.
  • Figure 2: Some bubble diagrams. The first three are cactus diagrams.
  • Figure 3: Quartic polynomial $V_\alpha(\lambda)$ (magenta) and corresponding eigenvalue distribution $\rho_{\mathrm{ext}}(\lambda)$ (orange, dashed) for $\alpha=1$ (left), $\alpha=0$ (center), and $\alpha=\alpha_c$ (right).
  • Figure 4: Multicritical polynomial $V(\lambda)$ for $m= 3$ (magenta) and corresponding eigenvalue distribution $\rho_{\mathrm{ext}}(\lambda)$ (orange, dashed) for $\gamma= 1/48$ (left) and $\gamma= \gamma_c$ (right).
  • Figure 5: Propagator and quartic vertex.
  • ...and 14 more figures