JT/SYK Spectral Form Factors and Universality Windows

The spectral form factor packages pairwise energy correlations into a time-domain observable. A dip, ramp, and plateau can reveal random-matrix universality, but only after the symmetry sector, temperature, connected subtraction, smoothing or ensemble average, and finite-size time window are declared. No such shape alone proves a unique JT/SYK duality.

Required background. Random Matrices, Spectral Statistics, and Ensemble Questions supplies symmetry classes and unfolding; Spectral Statistics, Form Factors, and Late-Time Evidence supplies the general diagnostic.

Helpful background. Finite Size, Symmetry Sectors, and Scrambling False Positives gives finite-system controls; Finite-N Spectra, Recurrences, and the Late-Time Plateau explains the exact late-time ceiling.

Evidence cutoff: 25 July 2026.

Thermal and microcanonical definitions

For levels $E_n$ in one symmetry block,

$$Z(\beta+it)=\sum_n e^{-(\beta+it)E_n}, \qquad K(\beta,t)=\overline{\left|Z(\beta+it)\right|^2}.$$

The bar must be specified: disorder average, matrix-ensemble average, time smoothing, or an energy-window average are different operations. The connected form factor is

$$K_c(\beta,t) =K(\beta,t) -\overline{Z(\beta+it)}\, \overline{Z(\beta-it)}.$$

A microcanonical form factor instead selects levels with a window $w(E-E_0)$. It is preferable when the density and local mean spacing vary strongly across the canonical support. Normalizations such as $K/Z(\beta)^2$, $K/Z(2\beta)$, and the unnormalized $K$ answer different questions and must not be overlaid without conversion.

Why a ramp appears

Expanding the definition gives

$$K(\beta,t)= \sum_{m,n}e^{-\beta(E_m+E_n)}e^{-it(E_m-E_n)}.$$

The disconnected smooth density controls the early decay. Universal connected level correlations produce a ramp after the Thouless time $t_{\rm Th}$; diagonal terms produce a plateau near the Heisenberg time

$$t_H\sim \frac{2\pi}{\Delta E_{\rm local}}.$$

For the unitary sine kernel, Fourier transformation of the connected two-level correlation is linear over the corresponding bulk window and then saturates. The precise slope depends on normalization, density, and symmetry class—not merely on the word “ramp.”

First application: compare a sample with an ensemble

Compute $K_c$ for one finite-$N$ SYK realization and for an ensemble of realizations using the same symmetry block, inverse temperature, central energy, and window width. Report:

• the number of retained levels and samples;
• whether each spectrum was unfolded;
• the smoothing kernel and its width;
• $t_{\rm Th}$, $t_H$, and the fitted ramp interval;
• bootstrap uncertainty across realizations, not just time-bin scatter.

Cotler and collaborators found the characteristic long-time structure in SYK after suitable averaging Cotler et al. 2017. In JT gravity, the connected double-trumpet contribution reproduces the leading ramp of the associated matrix ensemble, while the nonperturbative spectrum supplies the plateau Saad, Shenker, and Stanford 2019, §§4–5. The two calculations share a universal correlation but perform different averages.

Adversarial controls

Remove the averaging from a small fixed sample. The result becomes a strongly fluctuating quasiperiodic signal, although a coarse ramp may re-emerge after time smoothing. Mix two symmetry sectors and the correlation hole and ramp change. Move the energy window toward a sparse edge and bulk scaling fails. If the conclusion disappears under all reasonable smoothing widths, it was not a stable universality claim; if it survives only after ensemble averaging, it remains an ensemble statement.

At fixed $N$, exact late-time behavior is determined by discrete gaps and recurrences. A classical JT saddle or perturbative topology sum cannot determine those sample-specific phases. Conversely, absence of a clean ramp in a tiny sample does not disprove local random-matrix correlations. The licensed conclusion is always tied to $N$, sector, window, average, and time interval.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Cotler, Jordan S., et al. “Black Holes and Random Matrices.” Journal of High Energy Physics 2017, 5 (2017): 118; erratum 2018, 9 (2018): 2. DOI. Open PDF.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.