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

Olivier Marchal (1) , Mattia Cafasso (2)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 CRM - Centre de Recherches Mathématiques [Montréal]

In this paper, we show that the double-scaling-limit correlation functions of a random matrix model when two cuts merge with degeneracy 2m are the same as the determinantal formulae defined by conformal (2m, 1) models. Our approach follows the one developed by Bergère and Eynard in (2009 arXiv:0909.0854) and uses a Lax pair representation of the conformal (2m, 1) models (giving a Painlevé II integrable hierarchy) as suggested by Bleher and Eynard in (2003 J. Phys. A: Math. Gen. 36 3085). In particular we define Baker-Akhiezer functions associated with the Lax pair in order 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.

Domaines: Physique mathématique [math-ph]

Olivier Marchal, Mattia Cafasso. Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case. Journal of Statistical Mechanics: Theory and Experiment, 2011, 2011, pp.P04013. ⟨10.1088/1742-5468/2011/04/P04013⟩. ⟨hal-00863578⟩