Random matrices with external source and multiple orthogonal polynomials. (English) Zbl 1082.15035

Bleher, P. M.; Kuijlaars, A. B. J.

Int. Math. Res. Not. 2004, No. 3, 109-129 (2004).

The authors consider a Hermitian random matrix ensemble of the form \[ \frac{1}{Z_n}\;e^{-\text{Tr}(V(M)-A,M)}\;dM. \] The fixed matrix \(A\) is called the “external source”. P. Zinn-Justin [Nucl. Phys., B 497, No. 3, 725–732 (1997; Zbl 0933.82022)] found a determinantal expression for the eigenvalue correlations in terms of a kernel \(K_n\). In the paper under review, the authors show that \[ K_n(x,y)=e^{-(1/2)(V(x)+V(y))}\;\sum_{k=0}^{n-1}P_k(x)Q_k(y) \] for some functions \(Q_k\), based on a characterization of the polynomial \[ P_n(z)=\mathbf{E}[\text{det}(z-M)]. \] The special form of the kernel in the case where \(A\) has precisely two eigenvalues is considered in detail.

Reviewer: Martín Argerami (Regina)

MSC:

15B52	Random matrices (algebraic aspects)
28C10	Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15A18	Eigenvalues, singular values, and eigenvectors

Keywords:

Hermitian random matrix; eigenvalue correlations

Citations:

Zbl 0933.82022