Spectral form factor and semi-circle law in the time direction

Abstract

We study the time derivative of the connected part of spectral form factor, which we call the slope of ramp, in Gaussian matrix model. We find a closed formula of the slope of ramp at finite $N$ with non-zero inverse temperature. Using this exact result, we confirm numerically that the slope of ramp exhibits a semi-circle law as a function of time.

15 pages, 4 figures; v2: minor corrections, references added

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• On the Spectral Form Factor for Random Matrices
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• Correlated Disorder in the SYK${}_2$ model
• Large $N$ expansion of an integrated correlator in $\mathcal{N}=4$ SYM
• Quantifying dip-ramp-plateau for the Laguerre unitary ensemble structure function
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Spectral Form Factor for Time-dependent Matrix model
• Exact 1/N expansion of Wilson loop correlators in $\mathcal{N}=4$ Super-Yang-Mills theory
• 't Hooft expansion of $SO(N)$ and $Sp(N)$ $\mathcal{N}=4$ SYM revisited
• Detecting few-body quantum chaos: out-of-time ordered correlators at saturation
• Replicas for Random Matrices
• The Phases of Chaos