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

We give a hyperpfaffian formulation for correlation functions in $\beta$-ensembles arising in random matrix theory and statistical mechanics when $\beta = L^2$ 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 $R_m : W^m \rightarrow [0, \infty)$ we associate the $L$-vector valued function $\gamma_m : W^m \rightarrow \Lambda^L \mathbb C^{L(M-m)}$ such that $R_m(\mathbf y)$ is given by the Vandermonde determinant in $y_1, \ldots, y_m$ times the hyperpfaffian of $\gamma_m(\mathbf y).$ The partition function of the ensemble was previously shown to be the hyperpfaffian of a {\it Gram} $L$-form $\gamma$ in $\Lambda^L \mathbb C^{LM},$ and we demonstrate the relationship between $\gamma_m(\mathbf y)$ and $\gamma$, 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 $\gamma(\mathbf y)$ so it is very sparse (relative to the expected ${L(M-m) \choose L}$ coefficients of a general $L$-vector). These generalize skew-orthogonal polynomials arising in the well-understood $\beta = 4$ situation. Finally we explore the situation in the circular $\beta = L^2$ ensembles. Here the monomials give a prototype, and we give explicit formulas for $\gamma$ and $\gamma_m$ in this setting. We use our hyperpfaffian framework to produce exact formulas for the two point function when $\beta = 16$ for small values $M.$ Along the way we will record hyperpfaffian evaluations using known values of partition functions of $\beta$-ensembles.