Maximum of the Characteristic Polynomial of Random Unitary Matrices
Maximum of the Characteristic Polynomial of Random Unitary Matrices
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DOI：
10.1007/s00220-016-2740-6
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发表时间：
2017-01-01
影响因子：
2.4
通讯作者：
Bourgade, Paul
中科院分区：
文献类型：
--
作者：
Arguin, Louis-Pierre;Belius, David;Bourgade, Paul
最近Fyodorov，Hiary和Keating证明了从Haar测度采样的随机酉矩阵的单位圆上的特征多项式的最大值的增长类似于随机变量C。在本文中，我们验证了该猜想的领先顺序，即我们证明了对于任意小的情况，最大值以很高的概率位于范围内。该方法基于在特征多项式的傅里叶分解中识别近似分支随机游走，并使用开发的技术来描述分支随机游走和其他对数相关随机场的极端情况。一个关键的技术投入是渐近分析的Toeplitz行列式与维数相关的符号。这些渐近的原始参数遵循的一般思想，1/f噪声随机能量模型的统计力学是由冻结过渡。我们还证明了随机酉矩阵的自由能的限制冻结。
It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a random unitary matrix sampled from the Haar measure grows like for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range , for arbitrarily small .The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols.The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/f-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.