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

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 ∼ x 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 [2] and uses a Lax pair representation of the conformal (2m, 1) models (giving Painlevé II integrable hierarchy) as suggested by Bleher and Eynard in [4]. 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.