Random Matrices, Spectral Statistics, and Ensemble Questions

Random-matrix theory predicts local correlations after a spectrum is restricted to one symmetry sector and unfolded to unit mean spacing. Agreement with those correlations is evidence for a universality class over a stated energy window. It does not identify a unique Hamiltonian, determine the smooth density of states, or prove a gravitational dual.

Required background. Matrix Eigenvalue Saddles, Loop Equations, and Phase Transitions supplies matrix-ensemble observables and large-$N$ densities.

Helpful background. Spectral Statistics, Form Factors, and Late-Time Evidence develops generic chaos diagnostics; Spectra, Resolvents, Spectral Measures, and Functional Calculus supplies spectral definitions; JT Topological Expansion and Weil–Petersson Volumes explains why a matrix integral appears in JT gravity.

Evidence cutoff: 25 July 2026.

Symmetry class before statistics

For a Hamiltonian block with no antiunitary symmetry, complex Hermitian matrices give the unitary class ($\beta_{\rm D}=2$). An antiunitary symmetry squaring to $+1$ gives the orthogonal class ($\beta_{\rm D}=1$); one squaring to $-1$ gives the symplectic class ($\beta_{\rm D}=4$) and Kramers pairs must first be treated correctly. Here $\beta_{\rm D}$ is the Dyson index, not inverse temperature.

All exact commuting charges, fermion parity, and discrete symmetries must be resolved before comparing levels. Superposing two independent symmetry blocks destroys level repulsion near zero spacing and can make a chaotic spectrum look Poissonian.

Unfolding and universal observables

Let $\overline N(E)$ be a smooth integrated density in a declared window. The unfolded levels

$$\xi_i=\overline N(E_i),\qquad s_i=\xi_{i+1}-\xi_i$$

have mean spacing one. For the unitary class, the bulk two-level cluster function approaches the sine-kernel result

$$R_{2,c}(s)=-\left(\frac{\sin\pi s}{\pi s}\right)^2 .$$

Nearest-neighbor spacing distributions and number variance test the same correlations at different ranges. The adjacent-gap ratio

$$r_i=\frac{\min(\delta_i,\delta_{i+1})} {\max(\delta_i,\delta_{i+1})}, \qquad \delta_i=E_{i+1}-E_i,$$

is locally scale-free and provides an important check on unfolding, though it still requires pure symmetry sectors and a stationary window.

First application: finite-N SYK level statistics

For a fixed SYK coupling realization:

1. choose $N$, $q$, and one irreducible symmetry block;
2. diagonalize that block and discard spectral edges unless edge statistics are intended;
3. estimate $\overline N(E)$ with several smoothing choices;
4. compare $P(s)$, $\langle r\rangle$, and the connected two-level function with the Dyson class predicted by the model’s antiunitary symmetry;
5. repeat across independent coupling realizations, reporting sample variance separately from level-window variance.

The $N\bmod 8$ dependence of Majorana SYK changes the antiunitary class, so a single Gaussian ensemble is not the correct target for all $N$. García-García and Verbaarschot demonstrated random-matrix correlations in the SYK spectrum with this symmetry dependence García-García and Verbaarschot 2016. Cotler and collaborators connected these correlations to longer-time spectral diagnostics Cotler et al. 2017.

Adversarial controls

Three deliberately bad analyses locate the inference boundary.

• Mix sectors. Combining independent parity or charge blocks creates spurious near-crossings and weakens repulsion.
• Overfit the unfolding. A high-order fit can subtract genuine long-range fluctuations; an underfit leaves the smooth density in the statistic. Stable conclusions survive several physically motivated windows and smoothers.
• Use the edge as bulk. The soft edge has Airy rather than sine-kernel scaling.

Passing these controls supports random-matrix universality in the tested sector and window. It does not specify which probability measure represents disorder, gravity topology, energy averaging, or an ensemble of microscopic theories. Exact finite-$N$ levels remain sample-specific.

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.
• García-García, Antonio M., and Jacobus J. M. Verbaarschot. “Spectral and Thermodynamic Properties of the Sachdev–Ye–Kitaev Model.” Physical Review D 94, 126010 (2016). DOI. Open PDF.