Explaining the 1/p^2 scaling of the ramp (Thouless) time in sparse SYK

Derive an analytic explanation for the empirically observed scaling t_ramp ≈ a/p^2 + b of the ramp (Thouless) time in the spectral form factor of the sparse q=4 Sachdev–Ye–Kitaev (SYK) model for p between p2 and p1, potentially via energy diffusion or a hydrodynamic description on the interaction hypergraph.

Background

The ramp time t_ramp marks the onset of universal random-matrix-theory behavior in the spectral form factor and is inversely related to the Thouless energy. The authors estimate t_ramp numerically using a threshold relative error method and find it nearly independent of p for p > p1, but fitting t_ramp ≈ a/p^2 + b for p between p2 and p1.

They explicitly note uncertainty about the origin of this 1/p^2 scaling and suggest it might be explained by energy diffusion on the sparse interaction graph or via an analytical framework inspired by sparse XXZ circuit models. A derivation would illuminate how sparsity controls the onset of universal chaos and strengthen connections to emergent hydrodynamics.