Random matrices with external source and multiple orthogonal polynomials

We show that the average characteristic polynomial P
n
(z) = E [det(zI−M)] of the random Hermitian matrix ensemble
$Z_n^{-1}\operatorname{exp}{(-\text{Tr}{(\text{V}{(\text{M})}-AM)})}dM$
is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a
j
of A, there is a weight, and P
n
has n
j
orthogonality conditions with respect to this weight, if n
j
is the multiplicity of a
j
. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin (1997). 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.