On Gaussian random matrices coupled to the discrete Laplacian

From superintegrability to tridiagonal representation of $β$-ensembles

“…In [1,2] the result (6) was deduced from some complicated combinatorics, which made it somewhat obscure and non-evident. The claim of the present paper is that one can simplify the problem by establishing the equivalence (1) = (6) for a complete set of polynomials F {p k }.…”

“…4 Comment on the derivation of (10) Constructing an elegant and universal way to prove superintegrability relations like (10) is an open problem since they have been explicitly formulated in [23] and derived in [28] from the Wick theorem. The fact that they can serve as an elementary substitute of the lengthy direct calculation in [1,2], can attract new attention to this problem, especially of experts in statistics, who know how to solve them in different ways. Still, at this moment the question remains open, and we make a choice in favor of a proof of (6) which is standard for string physicists, though somewhat transcendental from the point of view of the "down-to-Earth" subject like (10).…”

“…on the eigenvalues λ i of M = U † DU , with diagonal matrix D = diag(λ). Then one can integrate over the angular coordinates (unitary matrices) U and get an integral over eigenvalues λ only but with a non-trivial weight ∆(λ) 2 , made from the Vandermonde determinant ∆(λ) = N i<j (λ i −λ j ). One can further deform the power of the Vandermonde determinant in order to introduce β-ensemble…”

“…Among such approaches a very interesting one is an old observation [1,2] that the β-deformation looks technically simpler if one reduces the matrix integrals over M only partly: not to diagonal matrices, but to tridiagonal ones,…”

“…

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.

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