关于Fredholm行列式的数值计算
On the Numerical Evaluation of Fredholm Determinants
AI总结：
本文提出一种基于Nyström方法的通用数值方法计算Fredholm行列式，证明其误差与核的求积误差一致，并应用于随机矩阵理论中的分布函数和Airy过程的两点相关函数。
摘要：
数学和物理学中的一些重要量最自然地表示为积分算子的Fredholm行列式，特别是随机矩阵理论中的许多分布函数。尽管它们的数值值很有意义，但在文献中找不到对Fredholm行列式的系统数值处理。相反，少数可用的数值评估依赖于算子的特征函数展开（如果可以用特殊函数表示），或者依赖于数值上更直接的解析表达式（例如Painlevé超越函数），这些表达式在某些情况下被巧妙地推导出来。在本文中，我们通过研究投影方法，尤其是从经典的第二类Fredholm方程Nyström方法推导出的一种简单、易于实现、通用的Fredholm行列式数值计算方法，来填补文献中的空白。使用Gauss-Legendre或Clenshaw-Curtis作为底层求积规则，我们证明了近似误差本质上类似于核的截断的求积误差。特别地，对于随机矩阵理论中典型的解析核，我们得到指数收敛。详细讨论了该方法在Gaussian unitary ensemble (GUE)的分布函数（在体相和边缘标度极限下）中的应用。在将方法扩展到积分算子系统后，我们首次评估了最近研究的Airy和Airy1过程的两点相关函数。
英文摘要：
Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss-Legendre or Clenshaw-Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy1 processes for the first time.
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