The hitting time of nice factors

TL;DR

This paper extends the hitting time and threshold results from complete u-graphs to a broader class called nice u-graphs, using a process coupling approach and new combinatorial arguments.

Contribution

It generalizes the process coupling and hitting time results from complete u-graphs to nice u-graphs, and introduces new combinatorial bounds applicable to broader classes.

Findings

Hitting time results extended to nice u-graphs.

Process coupling method generalized for broader classes.

New combinatorial bounds developed for strictly 1-balanced u-graphs.

Abstract

Consider the random $u$-uniform hypergraph (or $u$-graph) process on $n$ vertices, where $n$ is divisible by $r>u\ge 2$. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete $u$-graph $K_r$, it also contains a $K_r$-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete $u$-graphs, but for a broader class of so-called nice $u$-graphs. The purpose of this article is to extend the process coupling for complete $u$-graphs to the full scope of the sharp threshold result: nice $u$-graphs. As a byproduct, we obtain the extension of the hitting time result to nice $u$-graphs. Since the relevant combinatorial bounds in the proof for the $K_r$-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the $K_r$-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations.