标题： 超pfaffian关联对于Beta-系综：Beta为偶平方整数 显示英文标题

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

摘要： 我们给出当$\beta = L^2$为偶数平方整数时，$\beta$系统中$M \times M$随机矩阵的相关函数的超pfaffian 表述。 更具体地说，对于第$m$个相关函数$R_m : \R^m \rightarrow [0, \infty)$，我们将其与$L$-向量值函数$\omega_m : \R^m \rightarrow \Lambda^L \R^{L(M-m)}$关联，使得$R_m(\mathbf y)$由$y_1, \ldots, y_M$中的范德蒙德行列式乘以$\omega_m.$的超pfaffian 给出。该系综的划分函数之前已被证明是{\it 格拉姆} $L$ -形式$\omega$在$\Lambda^L \R^{LM},$中的超pfaffian，我们展示了$\omega_m(\mathbf y)$与$\omega$之间的关系，两者系数均由单变量多项式的Wronskians 积分构建。 假设存在与系综权重相关的多项式族，我们可以构造$\omega(\mathbf y)$使其非常稀疏（相对于一般$L$向量的预期${L(M-m) \choose L}$系数）。 这些推广了在理解良好的$\beta = 4$情况中出现的斜正交多项式。 最后我们探讨圆盘$\beta = L^2$系综的情况。 在这里，单项式给出一个原型，我们给出了（圆形式的）$\omega$和$\omega_m.$的显式公式。我们使用超pfaffian 框架来当$\beta = 16$时为两点函数提供精确公式，对于小值$M.$。在过程中，我们将记录使用$\beta$系统的配分函数已知值的超pfaffian 计算结果。 显示英文摘要

摘要： We give a hyperpfaffian formulation for correlation functions in $\beta$-ensembles of $M \times M$ random matrices when $\beta = L^2$ is an even square integer. More specifically, to the $m$th correlation function $R_m : \R^m \rightarrow [0, \infty)$ we associate the $L$-vector valued function $\omega_m : \R^m \rightarrow \Lambda^L \R^{L(M-m)}$ such that $R_m(\mathbf y)$ is given by the Vandermonde determinant in $y_1, \ldots, y_M$ times the hyperpfaffian of $\omega_m.$ The partition function of the ensemble was previously shown to be the hyperpfaffian of a {\it Gram} $L$-form $\omega$ in $\Lambda^L \R^{LM},$ and we demonstrate the relationship between $\omega_m(\mathbf y)$ and $\omega$, 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 $\omega(\mathbf y)$ 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 (the circular versions of) $\omega$ and $\omega_m.$ 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.