School of Mathematical Sciences, Queen Mary University of London, , London E1 4NS, UK.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20120503. doi: 10.1098/rsta.2012.0503. Print 2014 Jan 28.
We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>N(x), x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line s = 1/2 + it of given constant length and present the results of numerical computations of the large values of ζ(1/2 + it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.
我们认为，在 1/f 噪声随机能量模型的统计力学中，先前推测的冻结转变情景，经过重新解释，控制了大 N×N 随机酉（圆形酉系综）矩阵 UN 的特征多项式 pN(θ)的模的最大值的概率分布；即当 N→∞时，pN(θ)的极值统计。此外，我们认为它导致了在相同极限下，在间隔 |pN(θ)|>N(x)，x>0 中，总长度 μN(x)的多重分形样行为。我们推测我们的结果扩展到黎曼 ζ 函数 ζ(s)在给定长度的临界线上 s = 1/2 + it 上的大值，并给出了 ζ(1/2 + it)的大值的数值计算结果。我们的主要目的是引起人们对这些不同极值问题之间意外联系的关注。
