Norm Bounds for Sparse Random Tensors and Spectral Gap of Random Hypergraphs

Friedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that $r$-uniform Erd\H{o}s-R\'enyi hypergraphs on $n$ vertices exhibit a spectral gap as soon as their expected number of hyperedges $m$ satisfies $m \gg n^{r/2}$. Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge. Our proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theorem of Talagrand (2021). As a consequence of our techniques, we obtain improved norm bounds for sparse random tensors with independent entries. Finally, under a mild moment equivalence assumption, we extend to tensors a seminal result of Seginer (2000) for random matrices with i.i.d. entries.