论文标题
经典的紧凑型组和高斯乘法混乱
The Classical Compact Groups and Gaussian Multiplicative Chaos
论文作者
论文摘要
我们将HAAR特征多项式的绝对价值的功率视为其论点的指数级别的随机矩阵以及指数的力量，这是单位圆的随机度量减去$ \ pm 1 $的小社区。我们表明，对于足够小的功率，在适当的归一化下，由于基质大小输入无穷大，这些随机度量的分布会融合到高斯乘法混乱。我们的结果类似于基督徒韦伯先前在[31]中建立的统一矩阵上的结果。因此，我们完成了经典的紧凑型组和高斯乘法混乱之间的联系。为了证明这一点，我们建立了适当的渐近公式，用于与奇异性合并的toeplitz和toeplitz+hankel的决定因素。使用Claeys等人最近传达给我们的公式，我们能够将结果扩展到整个单元圆。
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 minus small neighborhoods around $\pm 1$. 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 measure. Our result is analogous to one on unitary matrices previously established by Christian Webb in [31]. We thus complete the connection between the classical compact groups and Gaussian multiplicative chaos. To prove this we establish appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants with merging singularities. Using a recent formula communicated to us by Claeys et al., we are able to extend our result to the whole of the unit circle.
