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Bootstrapping the critical behavior of multi-matrix models

Masoud Khalkhali, Nathan Pagliaroli, Andrei Parfeni, Brayden Smith

TL;DR

The authors introduce bootstrapping with positivity to bound large-$N$ moments in multi-matrix models by coupling Schwinger-Dyson relations with positivity constraints from the Hamburger moment problem. They validate the approach on the quartic single-matrix model and apply it to several unsolved 2-matrix models, obtaining evidence for a string susceptibility exponent $\gamma=1/2$ in some cases (suggesting the Continuum Random Tree) and identifying other exponents that may signal new continuum limits. They also derive explicit first terms of the large-$N$ free energy as a power series in the coupling constants and study a 3-matrix model, demonstrating the method’s utility for extracting critical behavior and generating series expansions. The work broadens analytical access to multi-matrix models, offering a practical route to characterize phase structure and potential continuum limits, with potential extensions to tensor models and map enumerations."

Abstract

Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several multi-matrix models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $γ= 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.

Bootstrapping the critical behavior of multi-matrix models

TL;DR

The authors introduce bootstrapping with positivity to bound large- moments in multi-matrix models by coupling Schwinger-Dyson relations with positivity constraints from the Hamburger moment problem. They validate the approach on the quartic single-matrix model and apply it to several unsolved 2-matrix models, obtaining evidence for a string susceptibility exponent in some cases (suggesting the Continuum Random Tree) and identifying other exponents that may signal new continuum limits. They also derive explicit first terms of the large- free energy as a power series in the coupling constants and study a 3-matrix model, demonstrating the method’s utility for extracting critical behavior and generating series expansions. The work broadens analytical access to multi-matrix models, offering a practical route to characterize phase structure and potential continuum limits, with potential extensions to tensor models and map enumerations."

Abstract

Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several multi-matrix models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be , which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.
Paper Structure (32 sections, 1 theorem, 74 equations, 11 figures)

This paper contains 32 sections, 1 theorem, 74 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Outline of main results
  3. Preliminaries
  4. Matrix integrals
  5. The Schwinger-Dyson equations
  6. Bootstrapping with positivity
  7. The quartic model
  8. The 2-matrix models
  9. When $(g,\alpha,\beta) = (g,g,g)$
  10. Bootstrap bounds
  11. Series expansion at zero
  12. Critical behaviour
  13. When $(g,\alpha,\beta) = (g,-g,g)$
  14. When $(g,\alpha,\beta) = (g,g,-g)$
  15. When $(g,\alpha,\beta) = (-g,g,g)$
  16. ...and 17 more sections

Key Result

Proposition 1

The second moment $m_{2}$ of the matrix model eq:2-matrix model is such that for $g \in [-1/8,0)$. Additionally, and for $g\in (0,\infty)$.

Figures (11)

  • Figure 1: Bootstrapped solution of the quartic Hermitian matrix model for $g>0$. The colours correspond to the the size of the submatrix of the Hankel matrix as follows: Light blue is for 2 by 2, dark blue is for 3 by 3, green is for 4 by 4, and gold is for 5 by 5. The black curve is the analytic solution.
  • Figure 2: Bootstrapped solutions of the quartic Hermitian matrix model for $g <0$. The top left yellow region was computed with submatrices of size 6 by 6, the top right was computed with submatrices of size 7 by 7, and the bottom region was computed with submatrices of size 10 by 10. The critical point of this model is $-\frac{1}{12} = -.0833\overline{3}$.
  • Figure 3: Bootstrapped estimates of $m_{2}$ of the $(g,\alpha,\beta) = (g,g,g)$ 2-matrix model. Each region corresponds to the positivity constraints of various sizes of submatrices of the Hankel matrix \ref{['eq:Hankel 2 matrix']} overlaid as follows: yellow is 5 by 5, cyan is 9 by 9, and purple is 21 by 21. The subfigure on the right is only the latter constraint.
  • Figure 4: Bootstrapped estimates of $m_{2}$ of the $(g,\alpha,\beta) = (g,g,g)$ 2-matrix model for $g<0$. Each region corresponds to the positivity constraints of various sizes of submatrices of the Hankel matrix \ref{['eq:Hankel 2 matrix']} as follows: yellow is 5 by 5, cyan is 9 by 9, and purple is 21 by 21.
  • Figure 5: Bootstrapped estimates of $m_{2}$ of the $(g,\alpha,\beta) = (g,g,g)$ 2-matrix model for $g<0$. Each region corresponds to the positivity constraints of various sizes of submatrices of the Hankel matrix \ref{['eq:Hankel 2 matrix']} as follows: yellow is 5 by 5, cyan is 9 by 9, and purple is 21 by 21. This is compared with the estimates from the series expansion of the second moment in Section \ref{['sec:series expansion']}. Each coloured line from left to right represents successively more terms in expansion \ref{['eq: m2_expansion']}.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Conjecture 2
  • Conjecture 3