1202.1768 (Mario Kieburg)

Mixing of orthogonal and skew-orthogonal polynomials and its relation to
Wilson RMT    [PDF]

Mario Kieburg

The unitary Wilson random matrix theory is an interpolation between the
chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new
way of interpolation is also reflected in the orthogonal polynomials
corresponding to such a random matrix ensemble. Although the chiral Gaussian
unitary ensemble as well as the Gaussian unitary ensemble are associated to the
Dyson index $\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal
polynomials and skew-orthogonal polynomials. We consider the Hermitian as well
as the non-Hermitian Wilson random matrix and derive the corresponding
polynomials, their recursion relations, Christoffel-Darboux-like formulas,
Rodrigues formulas and representations as random matrix averages in a unifying
way. With help of these results we derive the unquenched $k$-point correlation
function of the Hermitian and then non-Hermitian Wilson random matrix in terms
of two flavour partition functions only. This representation is due to a
Pfaffian factorization drastically simplifying the expressions for numerical
applications. It also serves as a good starting point for studying the
Wilson-Dirac operator in the $\epsilon$-regime of lattice quantum
chromodynamics.

View original: http://arxiv.org/abs/1202.1768