Painlevé I double scaling limit in the cubic random matrix model

2013/10/14 by Pavel Bleher, Bleher, Pavel M., Alfredo Deaño +1
Mathematics · #05C10 #30E15 #33E17 #35Q15 #60B20 #Advanced Algebra and Geometry #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1310.3768

openalex publication_date 2013/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the N× 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 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.

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