2025/09/05 by Christopher D. Sinclair, Sinclair, Christopher D., Jonathan M. Wells +1
#math-ph #math.CO #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2509.05487
We give a hyperpfaffian formulation for correlation functions in β-ensembles arising in random matrix theory and statistical mechanics when β= L2 is an even square integer. More specifically, for ensembles of M points in a space W ⊂ \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 mth correlation function Rm : Wm → [0, ∞) we associate the L-vector valued function γm : Wm → ΛL \mathbb CL(M-m) such that Rm(\mathbf y) is given by the Vandermonde determinant in y1, …, ym times the hyperpfaffian of γm(\mathbf y). The partition function of the ensemble was previously shown to be the hyperpfaffian of a \it Gram L-form γ in ΛL \mathbb CLM, and we demonstrate the relationship between γm(\mathbf y) and γ, 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 γ(\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 β= 4 situation. Finally we explore the situation in the circular β= L2 ensembles. Here the monomials give a prototype, and we give explicit formulas for γ and γm in this setting. We use our hyperpfaffian framework to produce exact formulas for the two point function when β= 16 for small values M. Along the way we will record hyperpfaffian evaluations using known values of partition functions of β-ensembles.