Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case

TL;DR

This paper demonstrates that the double scaling limit correlation functions of a random matrix model with merging cuts are equivalent to those of conformal (2m,1) models, using a Lax pair and Painlevé II hierarchy.

Contribution

It establishes a connection between matrix model double scaling limits and conformal (2m,1) models via a Lax pair approach and Baker-Akhiezer functions.

Findings

Correlation functions match conformal (2m,1) models

Lax pair representation yields Painlevé II hierarchy

Constructs determinantal formulas for correlation functions

Abstract

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinantal formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.