[Paper Review] Janossy Densities I. Determinantal Ensembles

Alexei Borodin, Alexander Soshnikov|ArXiv.org|Dec 20, 2002

Random Matrices and Applications4 references8 citations

TL;DR

This paper derives an explicit formula for Janossy densities in determinantal point processes with finite-rank projection kernels, showing that these densities on an interval I are governed by the Christoffel-Darboux kernel of orthogonal polynomials on the complement of I. The key result establishes a direct link between Janossy densities and biorthogonal bases on the complement set, enabling exact computation of particle statistics in restricted regions for β=2 polynomial ensembles, including Laguerre and Jacobi ensembles.

ABSTRACT

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 can be expressed in terms of the Christoffel-Darboux kernel for the orthogonal polynomials on the complement of I.

Motivation & Objective

• To derive a general formula for Janossy densities in determinantal point processes with finite-rank projection-type kernels.
• To establish a connection between Janossy densities and orthogonal polynomials on the complement of a subset I in the underlying measure space.
• To provide explicit expressions for Janossy densities in β=2 polynomial ensembles, such as Laguerre and Jacobi ensembles.
• To enable exact computation of the distribution of the k-th smallest eigenvalue in random matrix ensembles via the derived kernel formula.

Proposed method

• The authors express the Janossy density kernel as $ L_I(x,y) = \sum_{j=1}^n \widetilde{\xi}_j(x) \widetilde{\eta}_j(y) $, where $ \widetilde{\xi}_j, \widetilde{\eta}_j $ are biorthogonal bases in $ L^2(X\setminus I, \mu) $.
• They use the identity $ L_I = K_I (Id - K_I)^{-1} $, where $ K_I $ is the restriction of the original kernel to the subset I.
• The method relies on biorthogonalization of the original functions $ \phi_j, \psi_j $ on the complement of I to construct the kernel $ L_I $.
• The approach applies to determinantal point processes with kernels derived from orthogonal polynomials, particularly in β=2 ensembles.
• The derivation uses Fredholm determinant identities and properties of orthogonal polynomials to relate Janossy densities to Christoffel-Darboux kernels.
• The method is validated through applications to Laguerre and Jacobi ensembles, yielding explicit asymptotic formulas for eigenvalue spacings.

Experimental results

Research questions

• RQ1 How can Janossy densities be explicitly computed for determinantal point processes with finite-rank kernels?
• RQ2 What is the relationship between Janossy densities on a subset I and orthogonal polynomials defined on the complement of I?
• RQ3 Can the distribution of the k-th smallest eigenvalue in β=2 polynomial ensembles be derived from the kernel structure of the Janossy density?
• RQ4 Is there a universal formula for Janossy densities in terms of biorthogonal bases on the complement set?
• RQ5 How do the asymptotic behaviors of Janossy densities in Laguerre and Jacobi ensembles relate to special functions like Bessel functions?

Key findings

• The Janossy density kernel $ L_I(x,y) $ is explicitly given by $ \sum_{j=1}^n \widetilde{\xi}_j(x) \widetilde{\eta}_j(y) $, where $ \widetilde{\xi}_j, \widetilde{\eta}_j $ are biorthogonal bases in $ L^2(X\setminus I, \mu) $, providing a constructive formula for the kernel.
• For β=2 polynomial ensembles, the Janossy density on an interval I is determined by the Christoffel-Darboux kernel of orthogonal polynomials on $ X\setminus I $, linking local statistics to global orthogonal structure.
• The probability that the smallest eigenvalue $ \lambda_1^{(n)} $ of the Laguerre ensemble with $ \alpha=0 $ exceeds $ s/n $ is exactly $ e^{-s} $, confirming a known result with a new derivation.
• The limiting distribution of $ \lambda_{k+1}^{(n)} $ as $ n \to \infty $ is given by $ e^{-s} \int_{(-s,0)^k} \det(K^{(0)}(x_i,x_j)) \, dx_1 \cdots dx_k $, with $ K^{(0)} $ the Bessel kernel.
• For the second smallest eigenvalue, the limit is $ \frac{e^{-s}}{2} \int_0^{2\sqrt{s}} x (I_0^2(x) - I_1^2(x)) \, dx $, expressed in terms of modified Bessel functions.
• The method applies to Jacobi ensembles as well, yielding analogous limit laws after appropriate rescaling of the interval I.