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arxiv: 2510.07804 · v4 · submitted 2025-10-09 · ✦ hep-th · quant-ph

Exploring the Spectral Edge in SYK Models

Bowen Ouyang , Pratik Rath This is my paper

Pith reviewed 2026-05-18 09:38 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords SYK modelspectral edgerandom matrix theoryquenched entropylevel spacing statisticssupersymmetric SYKJT gravityAiry model
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0 comments X p. Extension

The pith

The SYK model matches random matrix theory level spacing statistics near its spectral edge, reproducing the power-law behavior of quenched entropy at low temperatures.

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 SYK model behaves at the edge of its energy spectrum compared to expectations from random matrix theory. Numerical simulations reveal that the spacings between energy levels align with RMT ensembles even close to the edge. This alignment supports the prediction that the quenched entropy follows a power-law dependence on temperature at low values, avoiding the negativity seen in annealed entropy. Similar matching occurs in the supersymmetric SYK model for wormhole entanglement entropy.

Core claim

Numerical diagonalization shows that the level spacing statistics of the SYK Hamiltonian match the relevant RMT ensembles near the spectral edge. This leads to agreement with the RMT prediction for the power-law behaviour of the quenched entropy at low temperatures.

What carries the argument

The level spacing statistics near the spectral edge, which encode the continuous spectrum and determine the low-temperature thermodynamics of the quenched entropy.

If this is right

  • The quenched entropy in the SYK model follows the RMT-predicted power law at low temperatures.
  • In the N=2 supersymmetric SYK model, the quenched entanglement entropy of the wormhole can be extracted from BPS operator spectral properties.
  • Finite-size effects in SYK do not prevent matching to universal RMT behavior at the edge.

Where Pith is reading between the lines

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

  • This indicates that the extra structure in SYK does not disrupt the universal edge physics seen in pure RMT.
  • Similar numerical approaches could test edge behavior in other many-body chaotic systems.
  • The results support using SYK as a microscopic model for JT gravity including matter.

Load-bearing premise

That the numerical results from finite-N diagonalization accurately represent the universal properties of the continuous spectrum at the edge in the thermodynamic limit without corrections that change the power-law exponent.

What would settle it

Finding a different temperature power-law exponent for the quenched entropy in larger-scale simulations or analytic calculations would show that the level statistics do not fully match RMT at the edge.

read the original abstract

Previous work on Jackiw-Teitelboim (JT) gravity has shown that, at low temperatures, the annealed entropy becomes negative and departs from the quenched entropy. From the perspective of the random-matrix theory (RMT) dual of JT gravity, this effect is encoded in the continuous spectrum at the spectral edge that is universally described by the Airy model. At low temperature, the quenched entropy exhibits a power law dependence determined by the symmetry class of the RMT ensemble. Here we study the same question in the Sachdev-Ye-Kitaev (SYK) model which possesses much more structure than RMT. Through numerical simulations, we find that the level spacing statistics of the SYK model match the relevant RMT ensembles even near the spectral edge, thus leading to an agreement with the RMT prediction for the power-law behaviour of the quenched entropy at low temperatures. We also show similar effects in supersymmetric wormholes filled with matter, which is modeled by the $\mathcal N = 2$ supersymmetric SYK model. Numerically extracting the spectral edge properties of the BPS operators allows us to compute the quenched entanglement entropy of the wormhole in the large particle number limit.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript numerically studies the spectral edge of the SYK model and its N=2 supersymmetric extension. It reports that nearest-neighbor level spacing distributions extracted from exact diagonalization match the GOE/GUE predictions of random matrix theory even in energy windows near the edge where the density of states vanishes. This agreement is used to infer that the quenched entropy follows the RMT-predicted power-law decay at low temperature, and the same approach is applied to compute the quenched entanglement entropy of supersymmetric wormholes in the large-N limit.

Significance. If the finite-N results survive a controlled continuum limit, the work would strengthen the link between SYK models and the Airy-kernel RMT description of JT gravity, showing that universal edge statistics persist despite the additional structure of SYK. It would also provide a concrete route to extract low-temperature thermodynamics and entanglement quantities from spectral data in supersymmetric variants.

major comments (2)
  1. [Numerical results / level spacing analysis] Numerical results section (around the level-spacing histograms and entropy plots): the manuscript shows agreement between SYK nearest-neighbor spacings and RMT ensembles for N ≲ 32, but does not present an explicit extrapolation in N or in the scaled window width (measured in units of the local mean spacing). Because the DOS vanishes as a power law at the edge, the number of levels inside any fixed window drops rapidly; without demonstrating that the Kolmogorov-Smirnov distance to the RMT distribution vanishes as N → ∞ while the window is held at fixed multiple of the local spacing, the claimed universality remains vulnerable to finite-N or finite-window artifacts.
  2. [Quenched entropy at low temperature] Entropy extraction paragraph: the power-law exponent for the quenched entropy is read off from the RMT ensemble once level statistics are asserted to match. However, the manuscript does not quantify how sensitive the extracted exponent is to the precise choice of energy window or to the disorder averaging procedure; a small shift in the window can change the effective local spacing and therefore the inferred low-T scaling.
minor comments (2)
  1. [Abstract and introduction] The abstract states that the level spacing statistics 'match the relevant RMT ensembles'; it would help to specify in the main text which ensemble (GOE vs. GUE) is used for each SYK variant and to cite the exact RMT reference curves being compared.
  2. [Figures] Figure captions for the spacing distributions should include the precise energy window (in units of local mean spacing) and the number of disorder realizations used for each N.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our work exploring the spectral edge in SYK models. We address the major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: Numerical results section (around the level-spacing histograms and entropy plots): the manuscript shows agreement between SYK nearest-neighbor spacings and RMT ensembles for N ≲ 32, but does not present an explicit extrapolation in N or in the scaled window width (measured in units of the local mean spacing). Because the DOS vanishes as a power law at the edge, the number of levels inside any fixed window drops rapidly; without demonstrating that the Kolmogorov-Smirnov distance to the RMT distribution vanishes as N → ∞ while the window is held at fixed multiple of the local spacing, the claimed universality remains vulnerable to finite-N or finite-window artifacts.

    Authors: We agree that demonstrating convergence with N would strengthen the universality claim. Our study is constrained by the exponential growth of the Hilbert space, limiting exact diagonalization to N ≤ 32. We have ensured that windows are scaled to the local mean spacing, and the agreement holds across several such windows. In the revised manuscript, we will add a discussion of finite-N effects and include supplementary data on the Kolmogorov-Smirnov distance as a function of N for fixed scaled windows. revision: partial

  2. Referee: Entropy extraction paragraph: the power-law exponent for the quenched entropy is read off from the RMT ensemble once level statistics are asserted to match. However, the manuscript does not quantify how sensitive the extracted exponent is to the precise choice of energy window or to the disorder averaging procedure; a small shift in the window can change the effective local spacing and therefore the inferred low-T scaling.

    Authors: We have performed robustness checks by shifting the energy window and altering the disorder averaging. The power-law exponent for the quenched entropy shows minimal variation within the statistical uncertainties of our data. We will revise the manuscript to include a quantitative assessment of this sensitivity, such as a plot or table demonstrating the stability of the exponent under different window choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity: numerical comparison to external RMT

full rationale

The paper's central result is obtained via direct numerical diagonalization of finite-N SYK Hamiltonians, followed by extraction of nearest-neighbor level spacing distributions near the spectral edge and comparison to independent RMT ensemble predictions (GOE/GUE) taken from prior literature. This constitutes an external benchmark rather than any internal fitting, self-definition, or derivation that reduces to the paper's own inputs by construction. No load-bearing self-citations, ansatze smuggled via citation, or uniqueness theorems imported from the authors' prior work are invoked to establish the match; the agreement with the RMT power-law for quenched entropy follows from the observed statistics. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard domain assumptions from the SYK and JT gravity literature that the models share universal low-energy spectral features with RMT; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption SYK models share universal spectral edge properties with the Airy model of random matrix theory for the relevant symmetry class.
    Invoked when claiming agreement with RMT predictions for level spacing and quenched entropy.

pith-pipeline@v0.9.0 · 5731 in / 1439 out tokens · 40002 ms · 2026-05-18T09:38:37.228046+00:00 · methodology

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Reference graph

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