The classical compact groups and Gaussian multiplicative chaos
The classical compact groups and Gaussian multiplicative chaos
复制标题
经典紧群和高斯乘法混沌
作者：
J. Forkel;J. Keating
我们考虑Haar分布随机正交矩阵或辛矩阵的特征多项式的绝对值的幂，以及其参数的指数的幂，作为单位圆上的随机测度。我们还考虑了这些措施仅限于单位圆减去±1附近的小邻域的情况。我们证明了对于足够小的幂和在适当的归一化下，当矩阵大小趋于无穷时，这些随机测度在分布上收敛于高斯乘性混沌（GMC）测度。我们的结果类似于之前由Christian Webb建立的酉矩阵（2015 Electron）。20)。我们由此完成了经典紧群与GMC之间的联系。为了证明在排除±1附近的小邻域时的收敛性，我们建立了具有合并奇点的Toeplitz行列式和Toeplitz + Hankel行列式的适当渐近公式。使用Claeys等人（2021 Int.）的最新公式。数学。研究》。Rnaa354)，我们能够证明在整个单位圆上收敛。
We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a random measure on the unit circle. We also consider the case where these measures are restricted to the unit circle minus small neighborhoods around ±1. We show that for small enough powers and under suitable normalization, as the matrix size goes to infinity, these random measures converge in distribution to a Gaussian multiplicative chaos (GMC) measure. Our result is analogous to one relating to unitary matrices previously established by Christian Webb (2015 Electron. J. Probab. 20). We thus complete the connection between the classical compact groups and GMC. To prove this convergence when excluding small neighborhoods around ±1 we establish appropriate asymptotic formulae for Toeplitz and Toeplitz + Hankel determinants with merging singularities. Using a recent formula due to Claeys et al (2021 Int. Math. Res. Not. rnaa354), we are able to prove convergence on the whole of the unit circle.
影响因子：
3
作者：
Arguin, Louis-Pierre;Belius, David;Soundararajan, Kannan
通讯作者：
Soundararajan, Kannan