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

Abstract

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. We will prove a large deviation principle (LDP) for the normalized empirical measure of eigenvalues when qn = 2, in which case the eigenvalues can be expressed in term of these of Gaussian random antisymmetric matrices. Such LDP result has its own independent interest in random matrix theory. For general qn ≥ 3, we can not prove the LDP, we will prove a concentration of measure theorem by estimating the Lipschitz norm of the Gaussian SYK model.

Quantum resonant systems, integrable and chaotic

“…These systems are obtained from (1) by replacing the bosonic creationannihilation operators with their fermionic counterparts, removing the resonance condition, and treating the interaction coefficients C nmkl as random. This line of research has recently been given an extra boost through its conjectured connections to quantum black hole physics [32][33][34][35][36][37][38][39]. Here, a different aspect of technical similarity with our investigations emerges: while in our case, the Hamiltonian will turn out to be blockdiagonal with blocks of finite sizes due to the imposition of the resonant constraint, the fermionic version has a finite-dimensional space of states by construction.…”

Optimizing Strongly Interacting Fermionic Hamiltonians

“…A review from a high energy physics perspective is [Ros19]. The physics papers [GV16,GJV18] are rather accessible for those with a mathematics background, and the works [FTW19,FTW18,FTW20] give some entirely mathematical results.…”

“…Here we describe the heuristic developed in the physics community for predicting Opt(h) when h ∼ SYK q (n). We follow the discussion in [GJV18]; see also [FTW19,FTW18,FTW20] for partial formalizations of this heuristic.…”

Spectrum of SYK Model

“…In our subsequent papers [4,5], we further derive two theorems about the spectrum of the SYK model. The results are totally unknown in physics, but they are indeed the most fundamental and important theorems considered in random matrix theory.…”

“…These results imply some useful information about the (global) 2-point correlation of the eigenvalues. In [5], for the special case of the Gaussian SYK model, we will derive a large deviation principle for the normalized empirical measure of eigenvalues for q n = 2 (in which case it is a totally solvable system and physicists do not care it too much, but it does have its own interest in random matrix theory) and a concentration of measure theorem for general q n ≥ 3.…”