What is the Simplest Linear Ramp?

TL;DR

This paper explores how certain deterministic spectra, especially logarithmic ones related to the Riemann zeta function, can exhibit features like linear ramps in spectral form factors typically associated with random matrices, challenging conventional beliefs.

Contribution

It identifies conditions under which deterministic spectra display random matrix features, introduces the special role of logarithmic spectra, and connects spectral properties to the Riemann zeta function and black hole physics.

Findings

Logarithmic spectra produce linear ramps in spectral form factors.

Adding noise induces level repulsion in deterministic spectra.

Spectral form factor of log spectra relates to the Riemann zeta function.

Abstract

We discuss conditions under which a deterministic sequence of real numbers, interpreted as the set of eigenvalues of a Hamiltonian, can exhibit features usually associated to random matrix spectra. A key diagnostic is the spectral form factor (SFF) -- a linear ramp in the SFF is often viewed as a signature of random matrix behavior. Based on various explicit examples, we observe conditions for linear and power law ramps to arise in deterministic spectra. We note that a very simple spectrum with a linear ramp is $E_n \sim \log n$. Despite the presence of ramps, these sequences do $not$ exhibit conventional level repulsion, demonstrating that the lore about their concurrence needs refinement. However, when a small noise correction is added to the spectrum, they lead to clear level repulsion as well as the (linear) ramp. We note some remarkable features of logarithmic spectra, apart from their linear ramps: they are closely related to normal modes of black hole stretched horizons, and their partition function with argument $s=\beta+it$ is the Riemann zeta function $\zeta(s)$. An immediate consequence is that the spectral form factor is simply $\sim |\zeta(it)|^2$. Our observation that log spectra have a linear ramp, is closely related to the Lindel\"of hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through $|\zeta(it)|^2$ on a log-log plot is indeed $1$, to the fourth decimal. We also note that truncating the Riemann zeta function sum at a finite integer $N$ causes the would-be-eternal ramp to end on a plateau.