Title: Fisher-Hartwig Asymptotics and Log-Correlated Fields in Random Matrix Theory Show Chinese title

Title: 费舍尔-哈特维格渐近和随机矩阵理论中的对数相关场

Abstract: This thesis is based on joint work with Jon Keating [FK21], Tom Claeys and Jon Keating [CFK23], and Isao Sauzedde [FS22], and is concerned with establishing and studying connections between random matrices and log-correlated fields. This is done with the help of formulae, including some newly established ones, for the asymptotics of Toeplitz, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities. In Chapter 1, we give an introduction to the mathematical objects that we are interested in. In particular we explain the relations between the characteristic polynomial of random matrices, log-correlated fields, Gaussian multiplicative chaos, the moments of moments, and Toeplitz and Toeplitz+Hankel determinants with Fisher-Hartwig singularities. In Chapter 2 we use Riemann-Hilbert techniques to establish formulae for the asymptotics of Toeplitz, and Toeplitz+Hankel determinants with two complex conjugate pairs of merging Fisher-Hartwig singularities. In Chapter 3 we complete the connection between the classical compact groups and Gaussian multiplicative chaos, by showing that analogously to the case of the unitary group first established in [Web15], the characteristic polynomial of random orthogonal and symplectic matrices, when properly normalized, converges to a Gaussian multiplicative chaos measure on the unit circle. In Chapter 4 we compute the asymptotics of the moments of moments of random orthogonal and symplectic matrices, which can be expressed in terms of integrals over Toeplitz+Hankel determinants. The phase transitions we observe are in stark contrast to the ones proven for the unitary group in [Fah21]. In Chapter 5 we establish convergence in Sobolev spaces, of the logarithm of the characteristic polynomial of unitary Brownian motion to the Gaussian free field on the cylinder, thus proving the dynamical analogue of the classical stationary result in [HKO01]. Show Chinese abstract

Abstract: 本论文是与Jon Keating [FK21]、Tom Claeys和Jon Keating [CFK23]以及Isao Sauzedde [FS22]的联合工作，主要涉及建立和研究随机矩阵与对数相关场之间的联系。 这是通过使用公式来实现的，包括一些新建立的关于具有Fisher-Hartwig奇点的Toeplitz和Toeplitz+Hankel行列式的渐近性质的公式。 在第一章中，我们介绍了我们感兴趣的一些数学对象。 特别是我们解释了随机矩阵的特征多项式、对数相关场、高斯乘法混沌、矩的矩以及具有Fisher-Hartwig奇点的Toeplitz和Toeplitz+Hankel行列式之间的关系。 在第二章中，我们使用黎曼-希尔伯特技术，建立了具有两个复共轭合并的Fisher-Hartwig奇点的Toeplitz和Toeplitz+Hankel行列式的渐近性质的公式。 在第三章中，我们通过证明随机正交和辛矩阵的特征多项式在适当归一化后收敛到单位圆上的高斯乘法混沌测度，从而完成了经典紧群与高斯乘法混沌之间的联系，这类似于在[Web15]中首次建立的酉群的情况。 在第四章中，我们计算了随机正交和辛矩阵的矩的矩的渐近性质，这些可以表示为Toeplitz+Hankel行列式上的积分。 我们观察到的相变与[Fah21]中证明的酉群的相变形成鲜明对比。 在第五章中，我们建立了酉布朗运动的特征多项式的对数在Sobolev空间中的收敛性，从而证明了[HKO01]中经典平稳结果的动力学类比。