JT Spectral Statistics and Matrix Integrals

JT gravity has a perturbative sum over surfaces that is reproduced by a double-scaled random matrix integral. This is a precise relation between topology-weighted amplitudes and ensemble-averaged spectral correlators. It does not automatically turn every fixed holographic theory into an ensemble, nor does the genus expansion select a unique nonperturbative completion. The distinction matters most for late-time observables, connected multi-boundary amplitudes, and factorization.

Required background. Gauge Theory and String-Theory Preparation provides large-N and path-integral normalization conventions.

Helpful background. Quantum Gravity supplies the gravitational topology expansion; Quantum Information supplies spectral and ensemble language; Semiclassical Gravity provides Euclidean black-hole thermodynamics; and Black-Hole Information develops factorization and late-time questions.

Disk density and topology expansion

In a convenient rescaled JT convention, the leading disk partition function is a Laplace transform

$$Z_{0,1}(\beta)=\int_0^\infty dE\,\rho_0(E)e^{-\beta E}, \qquad \rho_0(E)=\frac{\sinh(2\pi\sqrt{E})}{4\pi^2}.$$

Here $E$ and $\beta$ are rescaled variables with $2C=1$. Restoring physical energy $E_{\rm phys}$ means $E=2CE_{\rm phys}$ and $\beta=\beta_{\rm phys}/(2C)$, including the Jacobian in the density; $e^{S_0}$ multiplies the disk density. For example, setting $E=x^2$ evaluates the displayed integral directly:

$$Z_{0,1}(\beta)=\frac{e^{\pi^2/\beta}}{4\sqrt\pi\,\beta^{3/2}}.$$

The exponential reproduces the classical Schwarzian saddle in these units; the power $\beta^{-3/2}$ is the soft-mode fluctuation factor. The exact Schwarzian measure is treated by Stanford and Witten 2017, §§ 2–3. Higher genus and additional boundaries organize as

$$\big\langle Z(\beta_1)\cdots Z(\beta_n)\big\rangle_c \sim\sum_{g=0}^{\infty}e^{(2-2g-n)S_0}Z_{g,n}(\beta_1,\ldots,\beta_n).$$

The brackets are essential: on the matrix side they denote an ensemble average. Saad, Shenker, and Stanford 2019, §§2–3.5, pp. 7–34 show that the Weil–Petersson recursion for the JT moduli-space integrals matches the matrix-model topological recursion. This is a term-by-term perturbative result in $e^{-S_0}$.

The disk density is smooth, whereas a single finite-dimensional Hamiltonian has a discrete density of delta functions. Fine spectral discreteness and the plateau therefore live beyond the naive genus-by-genus disk approximation. Smoothing, unfolding, and the order of the large-matrix and late-time limits must be declared before comparing spectral statistics.

Results and limitations through 2026-08-20

The controlling axes are topology, $e^{-S_0}$ order, matrix ensemble, spectral unfolding, time averaging, and fixed member versus ensemble average.

Evidence or position	Controlled result	Limitation or adverse discriminator
Saad, Shenker, and Stanford 2019, §§2–3.5, pp. 7–34	JT amplitudes at fixed genus and boundary number obey the same recursion as a double-scaled matrix integral	The asymptotic topology expansion alone does not define a unique nonperturbative contour or fixed Hamiltonian
Saad, Shenker, and Stanford 2019, §5, pp. 36–57	Matrix eigenvalue effects provide explicit exponentially small corrections and late spectral structure	Different nonperturbative choices can share the same perturbative genus series; some potentials have instabilities or contour ambiguities
Spectral form factor	After an ensemble or suitable time average, level repulsion yields a dip, ramp, and plateau	An unsmoothed fixed spectrum fluctuates strongly; averaging can conceal theory-specific recurrences and symmetries
Multi-boundary factorization	Connected wormhole amplitudes naturally compute ensemble-connected correlators	A fixed boundary theory requires exact factorization; identifying its product with an ensemble average leaves this requirement unresolved

The perturbative JT–matrix-integral relation is established and controlled. The competing positions concern interpretation: one treats the ensemble as fundamental to the bulk definition, another treats it as an effective average over microscopic theories, and a third seeks a fixed-theory completion whose coarse observables reproduce the same expansion. Evidence does not establish that these positions are equivalent.

Claims that the same matrix integral is therefore the unique microscopic completion of a particular fixed black-hole theory remain conjectural. The disputed step is not the recursion relation; it is the change from a controlled ensemble identity to an assertion about one ultraviolet Hilbert space. Keeping those logical levels separate preserves the genuine result while exposing the open question.

Ramp, plateau, and what is averaged

For a spectrum $E_i$, the spectral form factor is

$$g(\beta,t)=Z(\beta+it)Z(\beta-it) =\sum_{i,j}e^{-\beta(E_i+E_j)}e^{-it(E_i-E_j)}.$$

The disconnected part dominates early times. Correlated nearby levels generate a ramp after ensemble or time smoothing. For a discrete nondegenerate spectrum, the infinite-time average retains only $i=j$ and equals $Z(2\beta)$. With level degeneracies $d_E$, it is instead $\sum_E d_E^2e^{-2\beta E}$. Thus the height depends on normalization and exact symmetries; an unsmoothed single-spectrum signal continues to fluctuate. Matrix universality predicts local correlations once the symmetry class and unfolding are fixed, but does not determine every global spectral feature.

An independent check compares the disk spectral curve and the first connected two-boundary amplitude with the corresponding matrix resolvents. Another checks how the answer changes under a fixed-member calculation. If a quantity disappears without ensemble averaging, it cannot be advertised as an exact observable of a single theory unless a separate self-averaging argument is supplied.

Nonperturbative completion and factorization

An asymptotic series does not specify its exponentially small terms. The matrix integral supplies a possible completion after choosing a potential, contour, and eigenvalue prescription. Those choices can affect the extreme spectral tail while leaving every order of the genus expansion unchanged. “The matrix integral completion” is therefore too strong unless the nonperturbative data are stated.

The sharp adverse case is factorization. For a fixed theory on two disconnected boundaries, $Z_1Z_2$ factorizes exactly before any averaging. A connected Euclidean wormhole instead contributes naturally to an averaged connected correlator. Resolving this may require an ensemble interpretation, additional sectors, or a refined bulk observable. JT calculations demonstrate the tension; they do not by themselves decide the microscopic answer for a given higher-dimensional black hole.

What could change these conclusions

The conclusions could change if a theorem constructs a unique nonperturbative JT completion from specified boundary data, if an independent calculation derives the same late-time observables in a fixed Hamiltonian without hidden averaging, or if a counterexample changes the claimed matrix universality class. The conclusions also depend on whether a new contour, instability, exact factorization result, or measured spectral observable changes which exponentially small terms are allowed and which state class the matrix average represents.

Exercises

Explain why agreement of every coefficient in the fixed-time genus expansion does not by itself control the exact late-time spectral form factor. Which extra assumption gives the time-averaged plateau $Z(2\beta)$?

Solution

The expansion is taken at fixed time, whereas resolving a mean level spacing $\delta E$ requires times of order $1/\delta E$, which grow as $e^{S_0}$ in the relevant spectral region. The expansion need not be uniform in that limit. Beyond-all-orders ambiguities provide an additional reason it does not uniquely determine the exact spectrum; this does not mean that the plateau height is itself smaller than every perturbative term. For a discrete nondegenerate spectrum, time averaging the phase $e^{-it(E_i-E_j)}$ keeps only diagonal terms and gives $Z(2\beta)$. Exact degeneracies change that result to $\sum_Ed_E^2e^{-2\beta E}$. Comparing edge spectra or late-time correlators with the same averaging prescription can distinguish completions.

References

• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv preprint (2019). arXiv:1903.11115.
• Stanford, Douglas, and Edward Witten. “Fermionic Localization of the Schwarzian Theory.” Journal of High Energy Physics 10 (2017) 008. arXiv:1703.04612.