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Finite-N Spectra, Recurrences, and the Late-Time Plateau

A classical black-brane saddle replaces fine spectral lines by a smooth density and predicts quasinormal decay. A finite-volume, finite-NN quantum theory has a discrete spectrum: correlators are quasiperiodic, time-averaged squares remain nonzero, and sufficiently late recurrences occur. A ramp or plateau obtained after time or disorder averaging diagnoses specified spectral correlations; it does not by itself prove fixed-theory black-hole evaporation or reconstruct microscopic unitarity.

Required background. Corrections, Nonuniform Limits, and Failure Modes supplies the limit order; Ensemble Inequivalence and Microcanonical Saddles fixes the thermodynamic ensemble; Quasinormal Modes, Poles, and Spectral Response supplies the classical decay.

Helpful background. Shockwaves, OTOCs, and Scrambling supplies the earlier chaos window; Spectral Statistics, Form Factors, and Late-Time Evidence fixes spectral diagnostics; Nonperturbative Exponential Effects and Finite-N Sectors supplies invisible sectors.

Discrete spectral response

Section titled “Discrete spectral response”

For a thermal Hermitian operator,

CN(t)=1Z(β)∑m,ne−βEm∣Omn∣2ei(Em−En)t.C_N(t)=\frac{1}{Z(\beta)} \sum_{m,n}e^{-\beta E_m} \lvert O_{mn}\rvert^2e^{i(E_m-E_n)t}.

At finite NN this is an almost-periodic sum. Replacing the levels by a smooth density converts it into an integral whose phases can decay. That replacement is accurate at fixed time in a suitable large-NN limit but not uniformly to arbitrarily late time.

If the mean spacing in the relevant microcanonical band is δE\delta E, the Heisenberg time is

tH∼2πδE∼eS(E)Tt_H\sim\frac{2\pi}{\delta E} \sim \frac{e^{S(E)}}{T}

up to bandwidth and convention factors. At t∼tHt\sim t_H, individual levels are resolved. Poincaré recurrence times are generally much longer and are not fixed by this estimate.

Smooth saddle versus finite spectrum

Section titled “Smooth saddle versus finite spectrum”

Let a quasinormal approximation give Csaddle(t)∼e−ΓtC_{\mathrm{saddle}}(t)\sim e^{-\Gamma t}. A fine-grained contribution of typical size e−S/2e^{-S/2} becomes comparable when

e−Γtcross∼e−S/2,tcross∼S2Γ.e^{-\Gamma t_{\mathrm{cross}}}\sim e^{-S/2}, \qquad t_{\mathrm{cross}}\sim\frac{S}{2\Gamma}.

The precise exponential power depends on normalization and on whether CC, ∣C∣2\lvert C\rvert^2, or a spectral form factor is studied. This is why a time window and observable must accompany every late-time claim. The tension between semiclassical decay and a discrete boundary spectrum was emphasized by Maldacena 2003.

For the spectral form factor,

K(β,t)=∣Z(β+it)∣2=∑m,ne−β(Em+En)e−it(Em−En).K(\beta,t) =\left\lvert Z(\beta+it)\right\rvert^2 =\sum_{m,n}e^{-\beta(E_m+E_n)} e^{-it(E_m-E_n)}.

After an appropriate connected subtraction and smoothing, chaotic spectra can show a dip, ramp, and plateau. Random-matrix behavior in holographic examples was analyzed by Cotler et al. 2017.

Averaging adversary

Section titled “Averaging adversary”

Compute ⟨K(t)⟩disorder\langle K(t)\rangle_{\mathrm{disorder}} in an ensemble and observe a clean ramp. Now ask for K(t)K(t) in one fixed Hamiltonian. It fluctuates strongly; smoothing or a spectral window may reveal correlations, but the ensemble curve is not the same observable. Inferring microscopic unitary evaporation in one theory from the averaged plateau has changed both the question and the evidence.

Jackiw–Teitelboim gravity has a concrete matrix-integral completion of its topology expansion Saad, Shenker, and Stanford 2019. That model-specific result does not make every higher-dimensional holographic CFT an ensemble.

Noncommuting limits and evidence ceiling

Section titled “Noncommuting limits and evidence ceiling”

Schematically,

lim sup⁡t→∞lim⁡N→∞∣CN(t)∣≠lim⁡N→∞lim sup⁡t→∞∣CN(t)∣\limsup_{t\to\infty}\lim_{N\to\infty}\lvert C_N(t)\rvert \neq \lim_{N\to\infty}\limsup_{t\to\infty}\lvert C_N(t)\rvert

when the strict saddle decays but finite systems recur. Evidence cutoff: 25 July 2026. A classical horizon and its quasinormal poles establish controlled coarse-grained relaxation. Spectral plateaus establish properties of the declared fixed, time-averaged, or ensemble-averaged observable. Neither alone proves a universal microscopic resolution of black-hole unitarity.

Thermal and Nonequilibrium QFT owns spectral statistics; later chapters own JT ensembles, topology sums, and unitarity conclusions.

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

Section titled “References”
  • Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori; Polchinski, Joseph; Saad, Phil; Shenker, Stephen H.; Stanford, Douglas; Streicher, Alexandre; and Tezuka, Masaki. “Black Holes and Random Matrices.” Journal of High Energy Physics 2017, 118 (2017). doi:10.1007/JHEP05(2017)118.
  • Maldacena, Juan. “Eternal Black Holes in Anti-de Sitter.” Journal of High Energy Physics 2003, 021 (2003). doi:10.1088/1126-6708/2003/04/021.
  • Saad, Phil; Shenker, Stephen H.; and Stanford, Douglas. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv:1903.11115.