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What is the Simplest Linear Ramp?
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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 $E_n \sim \log n$. 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.
Forward citations
Cited by 6 Pith papers
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A geodesic Witten diagram built directly in Euclidean BTZ coordinates reproduces the semi-classical Virasoro block and links the half-thermal-period timescale to the probe geodesic reaching the horizon.
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An Analytic Zeta Function Ramp at the Black Hole Thouless Time
The log-spectrum SFF equals |ζ(β+it)|^2; after removing the dip the β=0 ramp is exactly linear with slope 1 and time-average (π/24)t, giving an O(1) Thouless time.
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The moments of the spectral form factor in SYK
SYK spectral form factor moments match random matrix statistics at low order, with a k^2/N^{q-2} correction from spectral edge fluctuations that is amplified by sparsification.
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Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity
In the brickwall model of a BTZ black hole, hand-tuned Gaussian randomness at a stretched horizon reproduces random-matrix-theory spectral statistics and Krylov complexity peaks for scalar and fermionic probes.
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Blackish Holes
Placing a Dirichlet wall just outside the BTZ horizon produces a dense spectrum of normal modes that, in the continuum limit, yields a thermal two-point function at the Hawking temperature.
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