Spectrum of SYK Model

Abstract

This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.

An introduction to the SYK model

“…Finally, there are a number of topics which we have not discussed, such as: experimental realizations and quantum simulations of SYK [133][134][135][136][137][138][139], the zero-temperature entropy (an infinite N artifact) [140][141][142], studies of the spectral density, the spectral form factor, and connections with random matrix theory [83,[143][144][145][146][147][148][149][150][151][152][153].…”

Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs

“…In the moment method, this limit arises naturally in Wick contractions when treating all intersections as independent which gives approximate moments that are exact to order 1/N [9,34,39,55]. Remarkably, these moments turn out to give the spectral density of the weight function of the Q-Hermite polynomials with a nontrivial double scaling limit [9,39,55,56]. In this paper, when we speak of "corrections", we typically mean corrections to the Q-Hermite result.…”

Spectrum of SYK model III: Large deviations and concentration of measures

“…In this article, we will study the large deviation principle and the concentration of measure theorem for the Gaussian SYK model, instead of the general SYK model considered in [4,5].…”

“…Let λ i , 1 ≤ i ≤ L n be the eigenvalues of H. One may check that H is Hermitian by the anticommunitative relation (2), thus λ i are real numbers. One of the main tasks in random matrix theory is to understand the following normalized empirical measure of eigenvalues of H Let's first summarize the main results in [4,5]. Other than the standard Gaussian random variables, in [4], we consider the general cases where J i1i2···iq n are i.i.d.…”

“…

In [4], we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of law of large numbers in probability theory; in [5], we proved that the linear statistic of eigenvalues satisfies the central limit theorem. In this article, we continue to study another important theorem in probability theory -the concentration of measure theorem, especially for the Gaussian SYK model.

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