From superintegrability to tridiagonal representation of $β$-ensembles

Abstract

The wonderful formulas by I. Dumitriu and A.Edelman [1,2] rewrite β-ensemble, with eigenvalue integrals containing Vandermonde factors in the power 2β, through integrals over tridiagonal matrices, where βdependent are the powers of individual matrix elements, not their differences. These potentially useful formulas are usually deduced from rather complicated and non-transparent combinatorics and are not as widely known as they deserve. We explain that the superintegrability property, i.e. a simple expression of the Gaussian averages of arbitrary Jack polynomials through the same Jack polynomials, is immediately consistent with this tridiagonal representation, which may serve as a clue to its simple and transparent interpretation. For a formal non-perturbative proof, we use the Virasoro constraints, which themselves acquire an interesting structure in the tridiagonal realization. We also attract attention to the surprising spontaneous breakdown of discrete invariance by the tridiagonal measure, which may signal a new interesting anomaly at the elementary level of the basic eigenvalue matrix model.

Superintegrability in $β$-deformed Gaussian Hermitian matrix model from $W$-operators

“…This deformation is introduced in the form of integral of eigenvalues (for a matrix integral representation of this model see [26,27]). In general one can consider integrals not only of Hermitian matrices but of orthogonal or simplectic.…”

Superintegrability in $$\beta $$-deformed Gaussian Hermitian matrix model from W-operators

“…Namely, we prove the superintegrability of the βdeformation of the Gaussian Hermitian matrix model. This deformation is introduced in the form of integral of eigenvalues (for a matrix integral representation of this model see [26,27]). In general one can consider integrals not only of Hermitian matrices but of orthogonal or simplectic.…”