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arXiv:2608.27627

The Late-time Ramp of the Double-Scaled SYK Model from the large-nnn tail of Cactus Diagram

Yao Li

Published Aug 27, 2026• 0 citations
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Abstract

We derive the late-time ramp of the finite-temperature spectral form factor in the double-scaled SYK model directly from cactus diagrams, which is introduced as a multi-trace generalization of chord diagram in Ref \cite{Berkooz:2020fvm}. Although single-trace observables admit an exact chord-diagram description, multi-trace sums are obstructed by the chord-intersection weights $q_{IJ}$, which cannot be averaged independently to $q$. Our key idea is that the non-analytic contribution responsible for the ramp is controlled by the large-order tail of the Cactus-diagram expansion and is insensitive to finite changes in its low-order analytic terms. Decomposing the contribution with $n$-cross-trace-pairings Cactus diagram into two buds $B_n$ and a kernel $\mathcal K_n$, we determine their large-$n$ asymptotics with fixed $0\leq q<1$ and resum the resulting tail. For $β_L=β+it$ and $β_R=β-it$, we obtain \begin{equation*} Z_s^{\mathrm{sing}}(β+it,β-it) =s_p c_N \frac{|t|}{2π} \int_{E_{min}}^{E_{max}} e^{-2βE} dE+O(1), \qquad s_p=\begin{cases}2,&4\mid p,\\1,&4\nmid p.\end{cases} \end{equation*} This result reproduces the linear ramp predicted by random matrix theory, including its temperature dependence and symmetry factor, and agrees with the semiclassical predictions. It provides a direct microscopic origin of the ramp within the general $q$-deformed quantum algebra. While the plateau lies beyond the scope of the present analysis and calculation.

DOI
10.48550/arxiv.2608.27627

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Abstract In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.
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A bstract We study the time evolution governed by the two-sided chord Hamiltonian in the double-scaled SYK model, which induces a probability distribution over operators in the double-scaled algebra. Through the bulk-to-boundary map, this distribution translates into dynamical profiles of bulk states within the chord Hilbert space. We derive analytic expressions for such profiles, valid across a broad parameter range and all time scales. Additionally, we demonstrate how distinct semi-classical behaviors emerge by localizing within specific energy regions in the semi-classical limit. We revisit the doubled Hilbert space formalism as an isometric map between the one-particle sector of the chord Hilbert space and the doubled zero-particle sector. Utilizing this map, we obtain analytic results for correlation functions and investigate the dynamical evolution for chord operators. Specifically, we establish an equivalence between the chord number generating function in presence of matter chords and the crossed four-point correlation function, the latter is closely related to the 6 j -symbol of Uq(su(1,1)) {U}_{\sqrt{q}}\left(\mathfrak{su}\left(1,1\right)\right) Uq​​(su(1,1)) U q su 1 1 . We also explore finite-temperature effects, showing that operator spreading slows as temperature decreases. In the semi-classical limit, we perform a saddle point analysis and incorporate the one-loop determinant to derive the normalized time-ordered four-point correlation function at infinite temperature. The leading correction reproduces the 1/ N connected contribution observed in the large- p SYK model. Finally, we examine the time evolution of total chord number in presence of matter in the triple-scaled regime, linking it to the renormalized two-sided length in JT gravity with matter.
The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.
Abstract In double-scaled SYK, the renormalized length of the Einstein–Rosen bridge is identified with ℓ = λn, where n is the chord number; at infinite temperature, n is the Krylov spread complexity of the thermofield double. We determine the late-time growth rate of ℓ, the limit of dℓ/dt as t → ∞, exactly, for any λ and any temperature. We prove that dℓ/dt converges to the group velocity 2λB sin θ of the asymptotic chord chain, averaged over the spectral measure of the state; the limit of ℓ/t follows. At infinite temperature this average had been obtained by stationary phase. The proof uses trace-class scattering theory and an odd extension of the free chain, and holds for every state with finite second moment of n. An exact Ehrenfest identity, d²ℓ/dt² = 2⟨q^n̂⟩ > 0, shows that the limit is approached from below at every time. Combined with the proof, it gives the speed of convergence: the residual decays as t⁻², with a closed-form coefficient fixed by the weight of the state at the two band edges, and as t^(−α/2−1) when the spectral density vanishes there as θ^α. At infinite temperature, the average is a convergent theta series, (2/π) Σₘ (−1)ᵐ q^(m(m+1)/2)/(1 − 4m²). It contains the known small-λ transseries and stays exact at large λ, where the two-term truncation errs by 42% at λ = 8. At any temperature, the same Fourier series turns the canonical average into a ratio of two convergent series of modified Bessel functions I₂ₘ(2βB). At finite temperature, the triple-scaling limit reproduces the canonical slope of the JT wormhole length obtained by Iliesiu, Mezei and Sárosi, and the leading finite-λ correction is a shift β → β + 2/(λB). At fixed λ, the canonical rate exceeds the classical value 2π/β for βJ ≳ 2π/√λ. The single-saddle estimate never does: at λ = 2 and βJ = 100 it underestimates the exact rate by a factor of six. Exact evolution of the thermofield double confirms the canonical formula at ten points, to within 9 × 10⁻⁴ at finite time and to 6 × 10⁻⁶ after extrapolation of the t⁻² residual; where the asymptotic regime is reached, the predicted coefficient of that residual agrees to 3%. At fixed entanglement entropy the rate still depends on the energy distribution of the state. The entanglement splits exactly into a mean microcanonical entropy and the spread of the distribution; the bridge grows with the first, so a broader distribution grows more slowly, within 4% of a simple law at λ = 0.05. An exact lower bound on the rate at fixed entanglement is derived in a separate paper.
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 N=2\mathcal{N}=2N=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 ℏ→0\hbar \to 0ℏ→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.