The Lenard Recursion Relation and a Family of Singularly Perturbed Matrix Models

TL;DR

This paper explores the connection between the Lenard recursion relation and a family of singularly perturbed matrix models, highlighting their relation to Painlevé equations and recent developments in double scaling limits.

Contribution

It introduces new links between the Lenard recursion formula and Painlevé III hierarchy within the context of singularly perturbed matrix models.

Findings

Connection established between Painlevé III hierarchy and Lenard recursion

New insights into double scaling limits of matrix models

Enhanced understanding of Painlevé equations in random matrix theory

Abstract

We review some aspects of recent work concerning double scaling limits of singularly perturbed hermitian random matrix models and their connection to Painlev\'{e} equations. We present new results showing how a Painlev\'{e} III hierarchy recently proposed by the author can be connected to the Lenard recursion formula used to construct the Painlev\'{e} I and II hierarchies.