arxiv: 2601.20954 · v2 · submitted 2026-01-28 · ✦ hep-th · gr-qc· quant-ph

Recognition: no theorem link

Spectral Form Factor of Gapped Random Matrix Systems

Krishan Saraswat

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:13 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph

keywords spectral form factorrandom matrix theorydegenerate ground statesChristoffel-Darboux kernelsine kernelJackiw-Teitelboim gravityBPS stateswormholes

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The pith

In random matrix models with many degenerate ground states and a macroscopic gap, the spectral form factor at low temperatures is dominated by the disconnected contribution at all times.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the spectral form factor behaves in random matrix ensembles that include a parametrically large number of degenerate ground states separated by a clear gap from the rest of the spectrum. It establishes that at sufficiently low temperatures the disconnected part overwhelms the connected part even at arbitrarily late times. The connected contribution is shown to depend only on the eigenvalues of the non-degenerate sector, which removes any effect from the degenerate states in calculations involving wormholes. Analysis via the Christoffel-Darboux kernel on concrete models such as the Bessel ensemble and N=2 Jackiw-Teitelboim supergravity reveals damped oscillations whose period is set by the gap size, together with a universal sine-kernel ramp whose slope matches the double-trumpet result.

Core claim

In random matrix models possessing a parametrically large number of degenerate ground states accompanied by a macroscopic gap, the spectral form factor at low temperature is dominated by the disconnected contribution for all times, while the connected form factor receives contributions solely from the non-degenerate eigenvalues and is governed by a sine-kernel ramp obtained from truncation of the Christoffel-Darboux kernel in the ħ→0 limit.

What carries the argument

The Christoffel-Darboux kernel whose truncation to the sine kernel in the ħ→0 limit supplies the connected ramp and isolates the non-degenerate sector.

If this is right

The connected ramp agrees with the leading double-trumpet geometry in the gravity dual.
Damped oscillations appear in the disconnected form factor with period inversely proportional to the gap size.
The time at which the connected ramp transitions to a plateau depends on the leading spectral density of states.
BPS states make no contribution to the connected spectral form factor and therefore play no role in wormhole calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In supersymmetric gravity models the late-time wormhole physics would be insensitive to the BPS sector.
The gap introduces a new oscillatory timescale visible in the disconnected form factor that could be measured in finite-temperature matrix-model simulations.
The predicted dependence of the plateau onset on the spectral density offers a direct way to test the truncation approximation against exact diagonalization of larger ensembles.

Load-bearing premise

The ensembles contain a parametrically large number of degenerate ground states separated by a macroscopic gap, and the Christoffel-Darboux kernel reduces to the sine kernel in the semiclassical limit.

What would settle it

A numerical computation of the connected spectral form factor in an explicit gapped matrix model at low temperature that shows a non-zero contribution from the degenerate eigenvalues or that deviates from the sine-kernel ramp would falsify the claim.

read the original abstract

In this work, we study the spectral form factor of random matrix models which exhibit a large number of degenerate ground states accompanied by a macroscopic gap in the spectrum. The central aim of this work is to understand how the standard narrative about the behavior of the spectral form factor is modified in the presence of these parametrically large number of ground states. We show that, at sufficiently low temperatures, the spectral form factor is dominated by the disconnected contribution, even at arbitrarily late times. Moreover, we demonstrate that the connected form factor only depends on the eigenvalues of the non-degenerate sector, implying that BPS states do not contribute to wormhole calculations in the gravity context. Using the Christoffel-Darboux kernel, we analyze a number of examples including the Bessel model and $\mathcal{N}=2$ Jackiw-Teitelboim supergravity. In these examples, we find damped oscillations in the disconnected form factor, with a period set by the inverse size of the gap. Furthermore, we demonstrate that the slope of the ramp in the connected form factor arises from a universal sine-kernel, which emerges from a truncation of the full non-perturbative kernel in the $\hbar \to 0$ limit, and find agreement with the leading double trumpet result. Finally, we present predictions for how the ramp will transition to a plateau in the connected form factor and demonstrate how the transition depends on the details of the leading spectral density of states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows disconnected SFF dominance at late times in gapped degenerate spectra and claims BPS states drop out of the connected part, but the sine-kernel truncation at the gap edge is the weak link.

read the letter

The main thing here is that in random matrix models with a macroscopic gap and lots of degenerate ground states, the disconnected spectral form factor takes over at low temperatures even at very late times. The connected piece only sees the non-degenerate eigenvalues, which would mean BPS states do not feed into wormhole saddles. They get this from the Christoffel-Darboux kernel applied to the Bessel model and N=2 JT supergravity, and they recover damped oscillations whose period tracks the gap size plus a ramp whose slope matches the double-trumpet result after a small-hbar truncation to the sine kernel. They also give predictions for how the ramp turns into a plateau depending on the leading density of states. That is the concrete new content. The calculations are straightforward once the kernel is in hand, and the agreement with existing gravity results is a clear plus. The soft spot is exactly the truncation step. Near the lower edge of a gapped spectrum the local statistics are governed by Airy or Bessel kernels rather than the bulk sine kernel, so it is not obvious the truncation remains uniform at late times. If the ramp slope or the decoupling of the degenerate sector shifts under a more careful treatment, the main claims about disconnected dominance and BPS exclusion would need revision. The abstract and the reported examples do not include explicit checks of the edge behavior or numerical tests of the limit, which leaves the central approximation under-supported. This is for people working on random-matrix models of quantum gravity and late-time holography. A reader who already knows the standard SFF story will see the modification clearly and can judge whether the kernel step holds up. It is worth sending to peer review because the regime is relevant and the methods are standard, but the referee should be asked to focus on whether the sine-kernel approximation survives at the gap edge.

Referee Report

3 major / 2 minor

Summary. The paper studies the spectral form factor (SFF) of random matrix models featuring a parametrically large number of degenerate ground states separated by a macroscopic gap. It claims that at sufficiently low temperatures the SFF is dominated by the disconnected contribution at all times, that the connected SFF depends only on the eigenvalues of the non-degenerate sector (implying BPS states decouple from wormhole contributions), and that the connected ramp arises from a universal sine kernel obtained by truncating the Christoffel-Darboux kernel in the ħ→0 limit. Explicit calculations are presented for the Bessel model and N=2 Jackiw-Teitelboim supergravity, yielding damped oscillations in the disconnected SFF (period set by the gap) and agreement with the leading double-trumpet result, together with predictions for the ramp-to-plateau transition that depend on the leading spectral density.

Significance. If the sector separation and kernel truncation are rigorously justified, the work supplies a concrete random-matrix derivation of the double-trumpet contribution in gapped systems with ground-state degeneracy and clarifies the decoupling of BPS states from gravitational wormholes. The explicit examples, damped-oscillation predictions, and density-dependent transition formulas add falsifiable content that can be checked against both matrix-model numerics and holographic calculations.

major comments (3)

[Kernel truncation analysis and connected SFF section] The central claim that the connected SFF is governed by the universal sine kernel (and therefore matches the double-trumpet result) rests on truncating the full non-perturbative Christoffel-Darboux kernel to the sine kernel in the ħ→0 limit. With a macroscopic gap, the lower edge of the non-degenerate spectrum lies at finite energy; local eigenvalue statistics near this edge are controlled by Airy or Bessel kernels rather than the bulk sine kernel. The manuscript must demonstrate that the truncation remains uniform at the late times relevant to the ramp, or else the slope and the conclusion that BPS states drop out may be altered. (Kernel truncation analysis and § on connected SFF)
[Derivation of sector separation for connected SFF] The assertion that the connected form factor depends only on the non-degenerate eigenvalues (and hence that degenerate ground states/BPS states do not contribute) requires an explicit derivation showing that the gap and degeneracy factor out of the connected two-point function. The current presentation leaves unclear whether this follows directly from the kernel or requires additional assumptions about the joint eigenvalue distribution across sectors.
[Transition predictions section] The prediction for the ramp-to-plateau transition in the connected SFF is stated to depend on the leading spectral density of states. The manuscript should supply the explicit functional form of this dependence (e.g., via the unfolded density or the Thouless time) and verify it against the Bessel and JT examples, as this is the only quantitative handle on how the plateau onset scales with gap size and degeneracy.

minor comments (2)

[Notation] Notation for the gap size and ground-state degeneracy should be introduced once and used consistently; the abstract and main text currently employ slightly different symbols.
[Figures] Figure captions for the numerical plots of the disconnected SFF should state the precise values of ħ, gap size, and degeneracy used, to allow direct comparison with the analytic damped-oscillation formula.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each of the major comments below and will incorporate the necessary revisions and clarifications into the updated version of the paper.

read point-by-point responses

Referee: The central claim that the connected SFF is governed by the universal sine kernel (and therefore matches the double-trumpet result) rests on truncating the full non-perturbative Christoffel-Darboux kernel to the sine kernel in the ħ→0 limit. With a macroscopic gap, the lower edge of the non-degenerate spectrum lies at finite energy; local eigenvalue statistics near this edge are controlled by Airy or Bessel kernels rather than the bulk sine kernel. The manuscript must demonstrate that the truncation remains uniform at the late times relevant to the ramp, or else the slope and the conclusion that BPS states drop out may be altered.

Authors: We thank the referee for highlighting this important subtlety. In our analysis, the ħ→0 limit is taken after unfolding in the bulk of the non-degenerate spectrum, where the local statistics are indeed described by the sine kernel for smooth densities. The macroscopic gap ensures that the relevant energy window for the ramp (at times t >> 1/gap) lies well within the bulk, away from the lower edge where Airy or Bessel kernels apply. Edge effects would only influence much earlier times or specific regimes not relevant to the universal ramp. We will revise the section on the connected SFF to include a more detailed discussion of the time scales and uniformity of the truncation, explicitly showing that the sine kernel governs the late-time ramp and that the BPS decoupling holds. revision: partial
Referee: The assertion that the connected form factor depends only on the non-degenerate eigenvalues (and hence that degenerate ground states/BPS states do not contribute) requires an explicit derivation showing that the gap and degeneracy factor out of the connected two-point function. The current presentation leaves unclear whether this follows directly from the kernel or requires additional assumptions about the joint eigenvalue distribution across sectors.

Authors: We agree that a clearer derivation is needed. The connected spectral form factor is the covariance of the partition function Z(β + it) and Z(β - it). Since the ground states are exactly degenerate and fixed at zero energy with no random fluctuations in their positions, they contribute only to the disconnected part. The joint probability distribution of eigenvalues separates into the degenerate sector (fixed) and the non-degenerate sector, so the connected two-point function is determined solely by the kernel of the non-degenerate eigenvalues. We will add an explicit derivation of this factorization in a new subsection, based directly on the definition of the Christoffel-Darboux kernel for the gapped ensemble. revision: yes
Referee: The prediction for the ramp-to-plateau transition in the connected SFF is stated to depend on the leading spectral density of states. The manuscript should supply the explicit functional form of this dependence (e.g., via the unfolded density or the Thouless time) and verify it against the Bessel and JT examples, as this is the only quantitative handle on how the plateau onset scales with gap size and degeneracy.

Authors: We will supply the explicit form in the revision. The ramp-to-plateau transition occurs at the Thouless time t_Th ≈ 2π / ρ(E_0), where ρ(E) is the macroscopic spectral density of the non-degenerate sector evaluated near its lower edge E_0 (the gap size), after appropriate unfolding. This is the standard result from RMT where the plateau sets in once the time resolution exceeds the mean level spacing 1/ρ(E). We will derive this dependence and verify it numerically/analytically against the Bessel model (with ρ(E) ∝ √E near the edge) and the N=2 JT supergravity example, providing the scaling with gap size and degeneracy. revision: yes

Circularity Check

0 steps flagged

No circularity: central claims derived from explicit kernel truncation on gapped ensembles

full rationale

The paper derives the dominance of the disconnected SFF at low temperature and the restriction of the connected SFF to the non-degenerate sector directly from the structure of the Christoffel-Darboux kernel applied to random-matrix ensembles possessing a macroscopic gap and parametrically many degenerate ground states. The sine-kernel ramp is obtained by an explicit truncation of the full non-perturbative kernel in the ħ → 0 limit, verified on concrete examples (Bessel model, N=2 JT supergravity). The reported agreement with the leading double-trumpet result functions as an external cross-check against gravity calculations rather than an input or self-referential premise. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivation remains self-contained against the stated RMT assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper rests on standard random-matrix modeling of chaotic spectra plus the new assumption of parametrically large ground-state degeneracy separated by a macroscopic gap; no new particles or forces are postulated.

free parameters (2)

gap size = macroscopic
Macroscopic gap size controls oscillation period and disconnected dominance; treated as a tunable parameter of the ensemble.
ground-state degeneracy = parametrically large
Parametrically large number of degenerate ground states is introduced to make the disconnected contribution dominant.

axioms (2)

domain assumption Random matrix ensembles with prescribed degeneracy and gap accurately capture the spectral statistics of the systems of interest.
Core modeling assumption invoked throughout the analysis of examples.
domain assumption The full non-perturbative kernel admits a truncation to the sine kernel in the ħ → 0 limit that yields the connected ramp.
Invoked to derive the universal ramp slope and match the double-trumpet result.

pith-pipeline@v0.9.0 · 5550 in / 1670 out tokens · 71338 ms · 2026-05-16T10:13:51.091129+00:00 · methodology

discussion (0)

Reference graph

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