Ergodicity of the Form Factor in Random Matrix Theory

Question

This question is motivated by recent works in quantum gravity, particularly in the analysis of the Sachdev-Ye-Kitaev (SYK) model. The SYK model is a one-dimensional quantum mechanical model which, in a certain limit, is dual to a two-dimensional black hole. (See this paper for reference.)

An interesting connection between black holes and random matrices (as discussed in the paper linked above) is obtained using the SYK model itself. The SYK model, because of its chaotic behaviour, allows ideas from random matrix theory to be invoked for its analysis. (The connection between quantum chaos and random matrix theory is pretty well-established. Quantum Signatures of Chaos by Fritz Haake serves as an excellent reference for this.)

One specific approach to this analysis is to look at the late-time behaviour of the spectral form factor for the model. The spectral-form-factor-versus-time plot exhibits an initial slope, an intermediate-time dip, and a late-time ramp and a final plateau. (See Section 3 of the above paper for details.) The paper argues that the late-time behaviour of the SYK model can be interpreted as random matrix behaviour, and in particular, discusses the physical (and mathematical) motivations of this behaviour.

On the other hand, there is the notion of the ergodicity of the form factor in random matrix theory as discussed in this paper (described specifically in Section VI). The paper proves the assertion that the form factor is ergodic in the sense that the time average becomes equivalent to the ensemble average in certain cases.

I've been trying to think about a possible connection between the late-time plateau in the spectral-form-factor-versus-time plot and the ergodicity of the form factor. It (naively) appears to me that the plateauing at large times can be understood in terms of the ergodicity of the form factor. Are there any known mathematical results that connect the two?

Answer 1

I don't think the late-time plateau in the spectral form factor is informative in the context of ergodicity of the ensemble of energy levels. The limit $K(t)\rightarrow 1$ for $t\rightarrow\infty$ and $N\rightarrow\infty$ seems a direct consequence of the central-limit theorem:

For $t>0$ the spectral form factor of a set of energy levels $E_1,E_2,\ldots E_N$ is given by $$K(t)=\frac{1}{N}\left|\sum_{n=1}^N e^{iE_nt/\hbar}\right|^2.$$ One can think of the sum $\sum_{n=1}^N e^{iE_nt/\hbar}$ as a random walk in the complex plane with steps of length 1 and direction angle $\phi_n=E_nt/\hbar$. For large $t$ and large $N$ the angles will form a dense set on the unit circle, so the random walk has a concentration of measure at a distance $\sqrt N$ from the origin. The late-time plateau $K(t)\rightarrow 1+{\cal O}(1/N)$ for $t\rightarrow\infty$ would then follow irrespective of the ensemble of energy levels.