Secular coefficients and the holomorphic multiplicative chaos
Secular coefficients and the holomorphic multiplicative chaos
复制标题
世俗系数和全纯乘法混沌
作者：
J. Najnudel;Elliot Paquette;N. Simm
我们研究了$N的长期系数 从循环矩阵U_{N}$中抽取N个随机酉矩阵U_{N}$。η $-ε，定义为特征多项式$det（1-zU_{N}^{*}）$中${z^n}$的系数。当$eta > 4$我们得到了一类新的极限分布，当$n$和$N$同时趋于无穷大时出现。我们解决了Diaconis和Gamburd的一个公开问题，证明对于$eta=2$，中间系数随着$N而趋于零 不需要钱。我们展示了如何高斯乘性混沌（GMC）理论在这些问题中发挥了突出的作用，并在得到的极限分布的明确描述。本文将Diaconis和Gamburd关于长期系数矩的显著幻方公式推广到所有的eta>0，并分析了矩的渐近性态。我们得到估计的数量级的长期系数的所有$eta > 0，$，当$预计2美元。这些见解促使我们引入一个新的随机对象与长期系数，我们称之为全纯乘法混沌（HMC）。把HMC看作一个随机分布，在适当的Sobolev空间中证明了它的正则性的一个尖锐结果。我们的证明暴露并利用了与其他领域的几个新的连接，包括随机排列，Tauberian定理和组合学。
We study the secular coefficients of $N imes N$ random unitary matrices $U_{N}$ drawn from the Circular $eta$-Ensemble, which are defined as the coefficients of ${z^n}$ in the characteristic polynomial $det(1-zU_{N}^{*})$. When $eta > 4$ we obtain a new class of limiting distributions that arise when both $n$ and $N$ tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd by showing that for $eta=2$, the middle coefficient tends to zero as $N o infty$. We show how the theory of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems and in the explicit description of the obtained limiting distributions. We extend the remarkable magic square formula of Diaconis and Gamburd for the moments of secular coefficients to all $eta>0$ and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all $eta > 0,$ and these estimates are sharp when $eta geq 2$. These insights motivated us to introduce a new stochastic object associated with the secular coefficients, which we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel connections with other areas, including random permutations, Tauberian theorems and combinatorics.
影响因子：
3
作者：
Arguin, Louis-Pierre;Belius, David;Soundararajan, Kannan
通讯作者：
Soundararajan, Kannan