论文标题
来自$ u（n）$的身份的Dyson Brownian Motion的Dip-Ramp-Plamp-Plateau
Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
论文作者
论文摘要
在最近的一项工作中，目前的作者表明，来自$ u（n）$的身份的dyson Brownian运动的特征值概率密度函数是新近识别的一类随机单位矩阵的示例，称为CyclicPólyaememembles。通常，后者表现出相关内核的结构化形式。专门针对$ u（n）$身份的戴森·布朗（Dyson Brownian）运动的情况允许光谱密度的力矩，而光谱符号因子$ s_n（k; t）$，可以根据某个高几何多项式进行明确评估。转换后，可以通过jacobi多项式$（n（μ-1），1）$确定这一点，其中$μ= k/n $和$ k $是整数标记傅立叶系数。从文献中的现有结果来看，可以指定光谱密度矩的渐近形式，可以指定$ \ lim_ {n \ to \ to \ infty} {1 \ over n} s_n（k; t; t）| _ {μ= k/n} $。这些反过来又使我们可以对平均$ \ langle的大$ n $行为进行定量描述| \ sum_ {l = 1}^n e^{i k x_l} |^2 \ rangle $。后者表现出倾斜 - 平移效应，从许多人体量子混乱的角度引起了最新的兴趣，并在黑洞中扰乱了信息。
In a recent work the present authors have shown that the eigenvalue probability density function for Dyson Brownian motion from the identity on $U(N)$ is an example of a newly identified class of random unitary matrices called cyclic Pólya ensembles. In general the latter exhibit a structured form of the correlation kernel. Specialising to the case of Dyson Brownian motion from the identity on $U(N)$ allows the moments of the spectral density, and the spectral form factor $S_N(k;t)$, to be evaluated explicitly in terms of a certain hypergeometric polynomial. Upon transformation, this can be identified in terms of a Jacobi polynomial with parameters $(N(μ- 1),1)$, where $μ= k/N$ and $k$ is the integer labelling the Fourier coefficients. From existing results in the literature for the asymptotics of the latter, the asymptotic forms of the moments of the spectral density can be specified, as can $\lim_{N \to \infty} {1 \over N} S_N(k;t) |_{μ= k/N}$. These in turn allow us to give a quantitative description of the large $N$ behaviour of the average $ \langle | \sum_{l=1}^N e^{ i k x_l} |^2 \rangle$. The latter exhibits a dip-ramp-plateau effect, which is attracting recent interest from the viewpoints of many body quantum chaos, and the scrambling of information in black holes.
