Painlev\'e I double scaling limit in the cubic random matrix model

Abstract

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the $N\times N$ Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of $N^{-2/5}$, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronqu\'ee solution to the Painlev\'e I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronqu\'ee 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.