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Asymptotic expansion of beta matrix models in the multi-cut regime

Gaëtan Borot, Alice Guionnet

Abstract

We establish the asymptotic expansion in $β$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling fractions of these segments are fixed, and show the existence of a $1/N$ expansion. We then study the asymptotics of the sum over the filling fractions, to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($β= 2$) as well as orthogonal ($β= 1$) and skew-orthogonal ($β= 4$) polynomials outside the bulk.

Asymptotic expansion of beta matrix models in the multi-cut regime

Abstract

We establish the asymptotic expansion in matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling fractions of these segments are fixed, and show the existence of a expansion. We then study the asymptotics of the sum over the filling fractions, to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model () as well as orthogonal () and skew-orthogonal () polynomials outside the bulk.

Paper Structure

This paper contains 79 sections, 46 theorems, 479 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Model and empirical measure
  4. Correlators
  5. Kernels
  6. Equilibrium measure and multi-cut regime
  7. Assumptions
  8. Main result with fixed filling fractions: partition function and correlators
  9. Relation with Chekhov--Eynard--Orantin topological recursion
  10. Main results in the multi-cut regime: partition function
  11. Comments relative to the geometry of the spectral curve
  12. Central limit theorems for fluctuations and their breakdown
  13. Asymptotic expansion of kernels and correlators
  14. Leading behaviour of the correlators
  15. Kernels
  16. ...and 64 more sections

Key Result

Theorem 1.1

Assume that $V\,:\, \mathsf{B} \rightarrow \mathbb{R}$ is a continuous function, and if $V$ depends on $N$, assume also that $V$ converges towards $V^{\{0\}}$ when $N$ goes to infinity in the space of continuous functions over $\mathsf{B}$ for the sup norm. Moreover, for $\tau \in \{\pm 1\}$ with $\ We consider the normalised empirical measure $L_N=N^{-1}\,M_N$ in the space $\mathcal{P}(\mathsf{B}

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.4
  • Theorem 1.8
  • Corollary 1.9
  • ...and 41 more