Generalized Christoffel–Darboux formula for classical skew-orthogonal polynomials

Abstract

We show that skew-orthogonal functions, defined with respect to Jacobi weight w a,b (x) = (1 − x) a (1 + x) b , a, b > −1, including the limiting cases of Laguerre (wa(x) = x a e −x , a > −1) and Gaussian weight (w(x) = e −x 2 ), satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2N × 2N matrices. Using the GCD formulae we calculate the level-densities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skew-orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices.

Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

“…There exist many works on the skew-Christoffel kernel for the specific weight cases (see e.g. [1,9]). Here we stress that the factorization (8) holds not only for the case mentioned above but also for any well-defined skew inner product.…”

“…This completes the proof. Now observing the discrete spectral transformation of even-degree SOPs (4), one can easily find that except for the multiplier factor, this is equivalent to the Christoffel-Darboux kernel for skew orthogonal polynomials (we will call this "skew-Christoffel kernel") [9,16] introduced in the theory of random matrices which takes the form…”

Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model

“…Skew-orthogonal polynomials are useful in the study of orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices [2,3,4,5,6,7,8,9,10,11,12,13]. In this paper, we derive asymptotics of skew orthogonal functions φ (β) m (x) and ψ (β) m (x) and their recursion coefficients [8,9], defined w.r.t.…”

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

“…together with the semi-infinite matrices describing the transitions between them. In the Gaussian, Laguerre, and Jacobi cases, the aforementioned recurrence relations involve three terms [155], so the resulting generalised Christoffel-Darboux formulae are relatively compact.…”