Hyperpfaffian Correlations for Beta-Ensembles: Beta an Even Square Integer

Abstract

We give a hyperpfaffian formulation for correlation functions in ${\hat{I}}^2$-ensembles arising in random matrix theory and statistical mechanics when ${\hat{I}}^2=L^2$ is an even square integer. More specifically, for ensembles of $M$ points in a space $W\subset \mathbb{C}$ (typically $W=\mathbb{R}$ or $W=\mathbb{T}$), arising either as eigenvalues of a random matrix or as a system of charged particles with log interaction, to the $m$th correlation function $R_m:W^m\to [0,\infty )$ we associate the $L$-vector valued function ${\hat{I}}_m^3:W^m\to \hat{I}{›}^L{\mathbb{C}}^{L(M-m)}$ such that $R_m(\mathbf{y})$ is given by the Vandermonde determinant in $y_1,\dots ,y_m$ times the hyperpfaffian of ${\hat{I}}_m^3(\mathbf{y}).$ The partition function of the ensemble was previously shown to be the hyperpfaffian of a {\it Gram} $L$-form ${\hat{I}}^3$ in $\hat{I}{›}^L{\mathbb{C}}^{LM},$ and we demonstrate the relationship between ${\hat{I}}_m^3(\mathbf{y})$ and ${\hat{I}}^3$, both having coefficients built from integrals of Wronskians of monic polynomials. Assuming the existence of families of polynomials sympathetic with the weight of the ensemble, we may construct ${\hat{I}}^3(\mathbf{y})$ so it is very sparse (relative to the expected $\left(\genfrac{}{}{0ex}{}{L(M-m)}{L}\right)$ coefficients of a general $L$-vector). These generalize skew-orthogonal polynomials arising in the well-understood ${\hat{I}}^2=4$ situation. Finally we explore the situation in the circular ${\hat{I}}^2=L^2$ ensembles. Here the monomials give a prototype, and we give explicit formulas for ${\hat{I}}^3$ and ${\hat{I}}_m^3$ in this setting. We use our hyperpfaffian framework to produce exact formulas for the two point function when ${\hat{I}}^2=16$ for small values $M.$ Along the way we will record hyperpfaffian evaluations using known values of partition functions of ${\hat{I}}^2$-ensembles.

20 pages, 5 figures; added a section on ${\hat{I}}^2=4$, minor revisions