基于仙人掌图大n尾部的双标SYK模型的后期斜坡
The Late-time Ramp of the Double-Scaled SYK Model from the large-$n$ tail of Cactus Diagram
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- Massachusetts Institute of Technology（麻省理工学院）
- Shanghai Jiao Tong University（上海交通大学）
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中文总结 AI 辅助
本研究从仙人掌图大n尾部推导双标SYK模型有限温度谱形式因子的后期斜坡，重现随机矩阵理论预测的线性斜坡，为斜坡提供了微观起源。
中文摘要 AI 辅助
我们直接从仙人掌图推导双标SYK模型中有限温度谱形式因子的后期斜坡，仙人掌图是文献\ucf1b{Berkooz:2020fvm}中引入的弦图的多迹推广。尽管单迹可观测量可采用精确的弦图描述，但多迹求和会被弦相交权重$q_{IJ}$阻碍，而$q_{IJ}$无法独立平均为$q$。我们的核心思路是，负责斜坡的非解析贡献由仙人掌图展开的大阶尾部控制，对其低阶解析项的有限变化不敏感。将具有$n$交叉迹配对的仙人掌图的贡献分解为两个芽$B_n$和一个核$\n\ucf1d{K}_n$，我们在固定$0\leq q<1$的情况下确定它们的大$n$渐近行为，并对所得尾部进行重求和。对于$\n\ucf1b{\beta}_L=\beta+it$和$\n\ucf1b{\beta}_R=\beta-it$，我们得到\n\n\n\ucf1b{Z}_s^{\mathrm{sing}}(\beta+it,\beta-it) =s_p c_N \frac{|t|}{2\pi} \int_{E_{min}}^{E_{max}} e^{-2\beta E} dE+O(1), \qquad s_p=\begin{cases}2,&4\mid p,\\\\1,&4\nmid p.\end{cases}\n\n该结果重现了随机矩阵理论预测的线性斜坡，包括其温度依赖性和对称因子，且与半经典预测一致。它为一般$q$变形量子代数内的斜坡提供了直接的微观起源，而平台超出了本分析和计算的范围。
英文摘要
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.
