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In many papers in Random Matrix Theory [1-3] related to quantum chaos (and, in particular, to the SYK model) they analytically continuate the partition function of the system $Z(\beta)$ into $Z(\beta + it)$ and then define the Spectral Form Factor like

\begin{equation} g(\beta,t)=\langle Z^*Z\rangle \end{equation}

They then claim that the specific shape of this function gives a lot of information about the level statistics of the system. Is there any paper or book where I can read a good introduction on these tools? Every paper I found doesn't explain anything in great detail and just shows graphs.

[1] J. Liu, "Spectral form factors and late time quantum chaos", Phys. Rev. D 98 086026 (2018), arXiv:1806.05316.

[2] A. Gaikwad and R. Sinha, "Spectral form factor in non-Gaussian random matrix theories", Phys. Rev. D 100 026017 (2019), arXiv:1706.07439.

[3] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, ... and M. Tezuka, "Black holes and random matrices", J. High Energ. Phys. 2017, 118 (2017), arXiv:1611.04650.

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The best paper I have found is arXiv:cond-mat/9608116. They use some nice techniques to represent the spectral form factor as a contour integral in the complex plane, and then analyse the saddle points for large N.

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There are many books that give an introduction to the theory of quantum chaos. I would suggest looking into the books of Hans-Jürgen Stöckmann: Quantum Chaos an Introduction and Fritz Haake: Quantum Signatures of Chaos. I would say, that these books should give you the information that you are asking for. In both books, the concept of SFF is introduced around page 110.

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