Double scaling limit in the random matrix model: The Riemann‐Hilbert approach

By Pavel Bleher, Alexander Its. 2002

172 citations recorded by the source corpus.

We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the ψ function for the Hastings‐McLeod solution to the Painlevé II equation. The proof is based on the Riemann‐Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points. © 2003 Wiley Periodicals, Inc.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f3006c05354ce3fd54d36d1821be56581af90890

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. Arno B. J. Kuijlaars
2. Pavel Bleher

In Mathematical Physics

1. Arno B. J. Kuijlaars
2. Pavel Bleher
3. M. Bertola
4. Alexander Tovbis
5. Alexander I. Aptekarev

In Statistics and Probability

1. Arno B. J. Kuijlaars
2. Pavel Bleher

Citations in the corpus

• Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach (2003, unknown): “See also [7], [ 8 ], [4], [23], [24], [25], [26], [27], [44] for recent developments.”
• Large n Limit of Gaussian Random Matrices with External Source, Part I (2004, methodology): “Our approach is based on the Riemann-Hilbert problem and it is applicable, in principle, to a general V . We develop the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert (RH) problems [18], thereby extending the works [3], [ 4 ], [16], [17], [25], [26] who treated the unitary invaria”
• Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0) (2007, unknown): “Certain critical cases of 2×2 matrix-valued Riemann-Hilbertproblems have been studied recently and it was found that the local Riemann-Hilbert analysis can be constructed in an explicit way with the isomonodromy characterization of certain Painlevé transcendents, see [15, 21, 22, 23, 24, 32, 43].”
• Random matrices with external source and multiple orthogonal polynomials (2003, methodology): “We expect that the Riemann-Hilbert problem for Y is tractable to asymptotic analysis using the methods of [3, 4] or [11, 12, 13].”
• Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I (2010, background): “• The case of random matrices where a spectral band splits into two (P2 equation [5] and hierarchy); • The trailing edge of the region of oscillations in the small–dispersion limit of KdV [9] (PII equation).”
• The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] (2001, unknown): “See [10, 13, 25] for an introduction, and [5, 9 , 17, 18, 19, 23, 26, 27, 28, 29, 30, 37, 41, 45, 46] for the latest developments.”