Reconciling sparse SYK Schwarzian emergence with linear-term scaling

Reconcile the large-N saddle-point claim of Anegawa, Iizuka, Mukherjee, Sake, and Trivedi (2023) that a Schwarzian mode described by Jackiw–Teitelboim gravity does not emerge in the sparse q=4 Sachdev–Ye–Kitaev (SYK) model when the number of nonzero interaction terms in the Hamiltonian scales linearly with N, with numerical evidence indicating that sparse SYK behaves like unsparsified SYK for fixed average interaction hypergraph degree k (i.e., a number of terms linear in N). Ascertain whether the threshold average degree k1 required for robust spectral rigidity and Schwarzian dynamics is N-independent or grows with N.

Background

The paper numerically studies the sparse SYK model (q=4) where each 4-fermion term is retained with probability p and rescaled to match energy scales, focusing on spectral form factor (SFF) and nearest-neighbor gap ratio r as chaos diagnostics. Two transitions are identified: p1 (above which the SFF and r match full SYK and RMT) and p2 (below which even nearest-neighbor statistics deviate), both scaling as 1/N^3 and parameterized by average degree k.

The authors’ findings suggest robustness of spectral rigidity and holographic features for sufficiently large constant k, while Anegawa et al. (2023) claim via a saddle-point analysis that the Schwarzian mode emerges only when the number of Hamiltonian terms is superlinear in N. The authors explicitly state uncertainty about reconciling these results and speculate that k1 might grow with N too slowly to detect. Clarifying this discrepancy would calibrate the sparse regime compatible with JT gravity and universal RMT behavior.