The Late-time Ramp of the Double-Scaled SYK Model from the large-n tail of Cactus Diagram
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 qIJ, 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 Bn and a kernel 𝒦n, we determine their large-n asymptotics with fixed 0≤ q<1 and resum the resulting tail. For βL=β+it and βR=β−it, we obtain Zssing(β+it,β−it) =sp cN |t|/2π ∫EminEmax e−2βE dE+O(1), sp=2,&4| p, 1,&4nmid p. 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.
Related
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BibTeX
@misc{li2026latetime,
title = {{The Late-time Ramp of the Double-Scaled SYK Model from the large-$n$ tail of Cactus Diagram}},
author = {Yao Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27627},
eprint = {2608.27627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27627},
}
RIS
TY - UNPB
TI - The Late-time Ramp of the Double-Scaled SYK Model from the large-n tail of Cactus Diagram
AU - Yao Li
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.27627
UR - https://arxiv.org/abs/2608.27627
AB - 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 𝒦_n, we determine their large-n asymptotics with fixed 0≤ q<1 and resum the resulting tail. For β_L=β+it and β_R=β−it, we obtain Z_s^sing(β+it,β−it) =s_p c_N |t|/2π ∫_E_min^E_max e^−2βE dE+O(1), s_p=2,&4| p, 1,&4nmid p. 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.
ER -