[Paper Review] Double scaling limit for matrix models with non analytic potentials

Mariya Shcherbina|arXiv (Cornell University)|Nov 7, 2005

Random Matrices and Applications17 references4 citations

TL;DR

This paper establishes the double scaling limit for unitary invariant random matrix ensembles with non-analytic potentials by deriving the asymptotic expansion of Jacobi matrix coefficients via perturbation theory of string equations. The leading-order terms are expressed through the Hastings-McLeod solution of the Painlevé II equation, and the limiting reproducing kernel is shown to satisfy a Dirac system with potential determined by these first-order terms, extending edge universality to non-analytic settings.

ABSTRACT

We study the double scaling limit for unitary invariant ensembles of random matrices with non analytic potentials and find the asymptotic expansion for the entries of the corresponding Jacobi matrix. Our approach is based on the perturbation expansion for the string equations. The first order perturbation terms of the Jacobi matrix coefficients are expressed through the Hastings-McLeod solution of the Painleve II equation. The limiting reproducing kernel is expressed in terms of solutions of the Dirac system of differential equations with a potential defined by the first order terms of the expansion.

Motivation & Objective

• To extend the double scaling limit analysis beyond analytic potentials in random matrix theory.
• To establish universality of local eigenvalue statistics in the edge regime for non-analytic potentials.
• To derive the asymptotic expansion of Jacobi matrix coefficients in the double scaling limit.
• To connect the limiting reproducing kernel to solutions of a Dirac system with potential from first-order perturbation terms.

Proposed method

• Perturbation expansion of the string equations governing the matrix model's spectral properties.
• Use of the Hastings-McLeod solution of the Painlevé II equation to express the first-order correction terms in the Jacobi matrix coefficients.
• Derivation of the limiting reproducing kernel as a solution to a Dirac system with potential defined by the first-order perturbation terms.
• Application of resolvent estimates and Combes-Thomas-type bounds to control matrix element decay in the Jacobi matrix.
• Use of Neumann series expansion for inverse Jacobi matrices to derive decay estimates for resolvent entries.
• Analysis of discrete difference equations to control growth and decay of perturbation terms in the Jacobi matrix entries.

Experimental results

Research questions

• RQ1 How does the double scaling limit behave for matrix models with non-analytic potentials?
• RQ2 Can the edge universality result—previously known for analytic potentials—be extended to non-analytic potentials?
• RQ3 What is the asymptotic structure of the Jacobi matrix coefficients in the double scaling limit for non-analytic potentials?
• RQ4 How is the limiting reproducing kernel related to solutions of a differential system in the non-analytic case?
• RQ5 What role does the Hastings-McLeod solution of Painlevé II play in the perturbative expansion of the Jacobi matrix?

Key findings

• The first-order perturbation terms of the Jacobi matrix coefficients are explicitly expressed in terms of the Hastings-McLeod solution of the Painlevé II equation.
• The limiting reproducing kernel of the matrix model is shown to satisfy a Dirac system of differential equations with a potential derived from the first-order perturbation terms.
• The double scaling limit is well-defined for non-analytic potentials under the given Hölder and logarithmic growth conditions.
• The asymptotic expansion of the Jacobi matrix coefficients is derived using perturbation theory of the string equations.
• The resolvent entries of the Jacobi matrix satisfy exponential decay estimates, consistent with Combes-Thomas-type bounds.
• The analysis confirms the existence of a positive constant δ satisfying the key inequality (2.52), ensuring the validity of the perturbative framework.