Necessary and sufficient conditions for melonic dominance in general hSYK models

Identify necessary and sufficient structural conditions under which a general, possibly dense deterministic hypergraph SYK model is melonic-dominated.

Background

The paper establishes melonic dominance for deterministic hypergraph SYK under three specific assumptions: one-overlap linearity, bounded local cycle rank, and a spectral gap in the line graph. These assumptions are sufficient for the large-degree analysis developed in the paper, but the authors do not characterize whether they are necessary or how melonic dominance behaves for more general, including dense, hypergraph constructions. Determining such a characterization would clarify the broader class of disorder-free SYK-like models that possess a melonic large-degree limit.