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

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 Bergère and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlevé 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.

37 pages, 4 figures. Presentation improved, typos corrected. Published in Journal Of Statistical Mechanics

References in corpus (4)

• The Hamiltonian Structure of the Second Painleve Hierarchy
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• Double Scaling Limit in Random Matrix Models and a Nonlinear Hierarchy of Differential Equations
• Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies

Cited by in corpus (4)

• Rational differential systems, loop equations, and application to the q-th reductions of KP
• Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas
• Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices
• Integrability-preserving regularizations of Laplacian Growth