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Matrix models at low temperature

Alice Guionnet, Édouard Maurel-Segala

TL;DR

The paper develops a rigorous framework for the low-temperature regime of Hermitian multi-matrix models with potentials $V_{\beta}=\beta U+W$, establishing trapping-based conditions that guarantee tightness of spectral distributions and that limit points satisfy Dyson-Schwinger equations. It then analyzes several regimes: (i) convex $U$ yields concentration near multiples of the identity with $\beta^{-1/2}$-type expansions; (ii) strong single-variable potentials produce spectral localization near finitely many minimizers with weights dictated by the local curvature; and (iii) the strong commutator model with $U=-[X,Y]^2$ leads to asymptotic commuting behavior with spectra concentrating at minimizers of $V_1,V_2$ and becoming independent, with explicit descriptions in quadratic cases. A unifying theme is the use of trapping/conf confinement, Dyson-Schwinger equations, and flow-based change-of-variables to obtain tightness, support localization, and asymptotic expansions, connecting to free-probability structures and planar-map enumerations in special limits. The results illuminate how energy-dominant potentials shape spectral localization and commutativity in the large-$N$ limit, with potential implications for free entropy, Yang–Mills toy models, and related non-commutative probability frameworks.

Abstract

In this article we investigate the behavior of multi-matrix unitary invariant models under a potential $V_β=βU+W$ when the inverse temperature $β$ becomes very large. We first prove, under mild hypothesis on the functionals $U,W$ that as soon at these potentials are "confining" at infinity, the sequence of spectral distribution of the matrices are tight when the dimension goes to infinity. Their limit points are solutions of Dyson-Schwinger's equations. Next we investigate a few specific models, most importantly the "strong single variable model" where $U$ is a sum of potentials in a single matrix and the "strong commutator model" where $U = -[X,Y]^2$.

Matrix models at low temperature

TL;DR

The paper develops a rigorous framework for the low-temperature regime of Hermitian multi-matrix models with potentials , establishing trapping-based conditions that guarantee tightness of spectral distributions and that limit points satisfy Dyson-Schwinger equations. It then analyzes several regimes: (i) convex yields concentration near multiples of the identity with -type expansions; (ii) strong single-variable potentials produce spectral localization near finitely many minimizers with weights dictated by the local curvature; and (iii) the strong commutator model with leads to asymptotic commuting behavior with spectra concentrating at minimizers of and becoming independent, with explicit descriptions in quadratic cases. A unifying theme is the use of trapping/conf confinement, Dyson-Schwinger equations, and flow-based change-of-variables to obtain tightness, support localization, and asymptotic expansions, connecting to free-probability structures and planar-map enumerations in special limits. The results illuminate how energy-dominant potentials shape spectral localization and commutativity in the large- limit, with potential implications for free entropy, Yang–Mills toy models, and related non-commutative probability frameworks.

Abstract

In this article we investigate the behavior of multi-matrix unitary invariant models under a potential when the inverse temperature becomes very large. We first prove, under mild hypothesis on the functionals that as soon at these potentials are "confining" at infinity, the sequence of spectral distribution of the matrices are tight when the dimension goes to infinity. Their limit points are solutions of Dyson-Schwinger's equations. Next we investigate a few specific models, most importantly the "strong single variable model" where is a sum of potentials in a single matrix and the "strong commutator model" where .
Paper Structure (16 sections, 32 theorems, 330 equations)

This paper contains 16 sections, 32 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Tightness of matrix models at finite temperature
  4. Proof of Theorem \ref{['conf-to-T']}
  5. Proof of Theorem \ref{['Ttosupport']}
  6. Asymptotic localization at low temperature
  7. Strong uniformly convex potentials
  8. Strong single variables potentials
  9. Proofs of Theorem \ref{['theosum']} and Corollary \ref{['corsum']}
  10. Low temperature expansion
  11. Strong commutator interaction
  12. Appendix
  13. Random matrix inequalities by flow
  14. Matrix inequalities
  15. Change of variables in one-matrix models
  16. ...and 1 more sections

Key Result

Theorem 2.2

Assume that there exists a positive real number $\eta$ and a finite real number $A$ such that $V$ in $\mathbb C\langle X_{1},\ldots, X_{\ell}\rangle^{c}_{sa}$ is $(\eta,A,I)$-trapping. Then with $L=32\max\left((1+|A| )d \eta^{-1},1\right)$, under $\mathbb Q^N_{V}$, almost surely Moreover, for every $\kappa\in (0,1)$, we have

Theorems & Definitions (60)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 4.1
  • ...and 50 more