Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$

TL;DR

This paper analyzes the spectral properties of Dyson Brownian motion on the unitary group, revealing explicit formulas for spectral moments and form factors, and describing the large N behavior exhibiting a dip-ramp-plateau pattern relevant to quantum chaos.

Contribution

It introduces a new class of random matrices called cyclic Pólya ensembles and derives explicit spectral density moments and form factors for Dyson Brownian motion from the identity on U(N).

Findings

Explicit formulas for spectral moments and form factors in terms of hypergeometric and Jacobi polynomials.

Asymptotic analysis of spectral density moments as N approaches infinity.

Identification of the dip-ramp-plateau behavior in spectral form factors relevant to quantum chaos.

Abstract

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 P\'olya 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 $(N(\mu - 1),1)$, where $\mu = k/N$ 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 $\lim_{N \to \infty} {1 \over N} S_N(k;t) |_{\mu = k/N}$. 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.