What is the Simplest Linear Ramp?

Das, S and Garg, SK and Krishnan, C and Kundu, A (2024) What is the Simplest Linear Ramp? In: Journal of High Energy Physics, 2024 (1).

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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 En ~ 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 = β + it is the Riemann zeta function ζ(s). An immediate consequence is that the spectral form factor is simply �ζ|(it)|2. Our observation that log spectra have a linear ramp, is closely related to the Lindelöf hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through |ζ(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. © 2024, The Author(s).

Item Type:	Journal Article
Publication:	Journal of High Energy Physics
Publisher:	Springer Science and Business Media Deutschland GmbH
Additional Information:	The copyright for this article belongs to Author.
Department/Centre:	Division of Physical & Mathematical Sciences > Centre for High Energy Physics
Date Deposited:	03 Mar 2024 07:44
Last Modified:	03 Mar 2024 07:44
URI:	https://eprints.iisc.ac.in/id/eprint/84203

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