论文标题
Sobolev空间中统一布朗运动特征多项式的对数的收敛
Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion in Sobolev space
论文作者
论文摘要
我们证明，随着矩阵尺寸流向无穷大，统一布朗运动的特征多项式对数的真实和虚构部分的收敛性在某些合适的Sobolev空间中保持，我们认为这是最佳的。这是Hughes，Keating和O'Connell [1]在固定时间内的结果的自然动力学类似物。自从Spohn [2]的工作被Bourgade和Falconet [3]广泛改善以来，已经知道了一种弱的收敛性。在这项研究的过程中，我们还证明了灯芯型身份，我们在本文中包括了它，因为它可能具有独立的兴趣。
We prove that the convergence of the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion toward Gaussian free fields on the cylinder, as the matrix dimension goes to infinity, holds in certain suitable Sobolev spaces, which we believe to be optimal. This is the natural dynamical analogue of the result for a fixed time by Hughes, Keating and O'Connell [1]. A weak kind of convergence is known since the work of Spohn [2], which was widely improved recently by Bourgade and Falconet [3]. In the course of this research we also proved a Wick-type identity, which we include in this paper, as it might be of independent interest.
