Random matrices with external source and multiple orthogonal polynomials
By Pavel Bleher, Arno B. J. Kuijlaars. 2003
182 citations recorded by the source corpus.
We show that the average characteristic polynomial Pn(z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z 1 n exp(−Tr(V (M) − AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue aj of A, there is a weight and Pn has nj orthogonality conditions with respect to this weight, if nj is the multiplicity of aj. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.
Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/f503e061e9c05a94d2eeddf9501e0cad002c1180
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In Applied Mathematics
- Arno B. J. Kuijlaars
- Pavel Bleher
- Walter Van Assche
- Franck Wielonsky
- Alexander I. Aptekarev
- Andrei Martínez-Finkelshtein
- K. Mclaughlin
- Maxim L. Yattselev
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Nikishin Systems Are Perfect (2010, unknown): “More recently, mixed type approximation has appeared in random matrix and non-intersecting brownian motions theories (see, for example, [6] and [13]).”
- Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, unknown): “[5], [8], [9], [11], [12], [13], [14], [27], [28], [29], [48] for again a very partial list.”
- Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach (2003, background): “Such larger-size Riemann‐ Hilbert problems have appeared recently in [ 9 ], [10], [14] in the context of random matrices with external sources.”
- Multiple orthogonal polynomial ensembles (2009, methodology): “The connection was first observed in [14] where MOPs were used in in a random matrix model with external source.”
- Large n Limit of Gaussian Random Matrices with External Source, Part I (2004, methodology): “Large n asymptotics of the multiple Hermite polynomials As noted in Section 2, see also [6], we have that the (1, 1) entry of the solution Y of the RH problem (1.”