Janossy densities. I: Determinantal ensembles. (English) Zbl 1070.15020

Borodin, Alexei; Soshnikov, Alexander

J. Stat. Phys. 113, No. 3-4, 595-610 (2003).

Summary: We derive an elementary formula for Janossy densities for determinantal point processes with a finite rank projection-type kernel. In particular, for \(\beta =2\) polynomial ensembles of random matrices we show that the Janossy densities on an interval \(I\subset {\mathbb R}\) can be expressed in terms of the Christoffel-Darboux kernel for the orthogonal polynomials on the complement of \(I\).
For Part II, see Zbl 1071.15027.

MSC:

15B52	Random matrices (algebraic aspects)
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
35Q15	Riemann-Hilbert problems in context of PDEs
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Keywords:

random matrices; orthogonal polynomials; Janossy densities; Riemann-Hilbert problem; determinantal point processes; finite rank projection-type kernel; Christoffel-Darboux kernel

Citations:

Zbl 1071.15027