Painlevé I double scaling limit in the cubic random matrix model

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the N×NN×N Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of N−2/5N−2/5, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronquée solution to the Painlevé I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronquée solution are limits of zeros of the partition function. The tools used include the Riemann&-Hilbert approach and the Deift&-Zhou nonlinear steepest descent method for the corresponding family of complex orthogonal polynomials and their recurrence coefficients, together with the Toda equation in the parameter space.The first author is supported in part by the National Science Foundation (NSF) Grants DMS-0969254 and DMS- 1265172. The second author acknowledges financial support from projects MTM2009–11686, from the Spanish Ministry of Science and Innovation, and from projects MTM2012–34787 and MTM2012-36732–C03–01, from the Spanish Ministry of Economy and Competitivity, and he is also grateful to the Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, for their hospitality in April–May 2012 and in August 2013, when a substantial part of this work was done. The research leading to these results has received financial support from the Fund for Scientific Research Flanders through Research Project G.0617.10 and from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement 247623–FP7-PEOPLE-2009-IRSES