Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit

Abstract

We consider the double scaling limit in the random matrix ensemble with an external source 1 Z n e −nTr( 1 2 M 2 −AM) dM defined on n × n Hermitian matrices, where A is a diagonal matrix with two eigenvalues ±a of equal multiplicities. The value a = 1 is critical since the eigenvalues of M accumulate as n → ∞ on two intervals for a > 1 and on one interval for 0 < a < 1. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case a = 1 new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Brézin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3 × 3-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

“…For recent progress related to discrete random walks, random tilings and random matrices with external source see [3,4,5,6,7,8,47,50].…”

Exact Solution of the Six‐Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase

“…These estimates of smallness of (ṽ R (z) − I) on Γ R enable us to solve the Riemann-Hilbert problem forR(z) by a series of perturbation theory. The fact that the radius ofŨ δ,k , r = δ k , is tending to zero does not cause a problem, see appendix to the work [5] of Bleher and Kuijlaars.…”

Analysis of the Large-n Limit of Multiple Orthogonal Polynomials and Non-intersecting Brownian Motions

“…More recently, it has been found [3]- [4] that multiple orthogonal polynomials are connected to the Gaussian Hermitian matrix model with an external source term AM, where/I is a fixed diagonal nxn real matrix. The partition function of this model is given by…”

“…for which (4) admits a unique solution /(s, z) satisfying (5) will be denoted by T. The solution /(s, z) will be referred to as the fundamental solution of the RH problem (4).…”

“…Moreover, the Gaussian matrix models with an external source (2) are related to the multiple orthogonal polynomials respect to the Gaussian weights (3), [3]- [4]. In this case the quasiclassical limit is characterized by e = l/« and h=l, and the Orlov functions are related to the joint probability density for the eigenvalues of M.…”