论文标题
一种随机统一矩阵的特征多项式的新方法
A new approach to the characteristic polynomial of a random unitary matrix
论文作者
论文摘要
自从Keating和Snaith的开创性工作以来，随机HAAR分布的单一基质的特征多项式已经看到了其几个功能性研究或变成了猜想。例如： $ \ bullet $它的值$ 1 $（Keating-Snaith Theorem）， $ \ bullet $其傅立叶系列的截断达到其学位的任何一部分， $ \ bullet $计算Birkhoff Polytope的相对体积， $ \ bullet $其产品和比率以不同的点为 $ \ bullet $其迭代衍生物在不同点上的产品， $ \ bullet $函数与$ \ mathbb {f} _q [x] $中的Divisor函数的总和有关。 $ \ bullet $它的中部系数， $ \ bullet $“时刻”，等等。 我们首次重新审视或计算这些最后功能和其他几个功能的整数矩的渐近学。我们使用的方法是一种基于再现核的非常通用的方法，这是对某些经典正交多项式的对称函数的概括，该函数被解释为特定随机变量的傅立叶变换以及这些随机变量的局部中央限制定理。此外，我们基于中位系数的随机分析提供了等效范式，以重新启动它们。这些方法为所有考虑的限制提供了一个新的统一框架，并在限制功能中解释了Hankel决定因素或Wronskians的幻影。
Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$ (Keating-Snaith theorem), $ \bullet $ the truncation of its Fourier series up to any fraction of its degree, $ \bullet $ the computation of the relative volume of the Birkhoff polytope, $ \bullet $ its products and ratios taken in different points, $ \bullet $ the product of its iterated derivatives in different points, $ \bullet $ functionals in relation with sums of divisor functions in $ \mathbb{F}_q[X] $. $ \bullet $ its mid-secular coefficients, $ \bullet $ the "moments of moments", etc. We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional.
