Kahn–Kim variance conjecture for random matchings in regular linear hypergraphs

Let $k\geq 2$ be an integer, let $H$ be an $n$-vertex $d$-regular linear $k$-graph, and let $M$ be a matching chosen uniformly from the set of all matchings of $H$. For a vertex $v$, write $\mathcal{P}_H(\overline{v})$ for the probability that $M$ does not cover $v$. Kahn and Kim's conjecture. For every $v\in V(H)$,

$$\mathcal{P}_H(\overline{v})=(1+o_d(1))d^{-1/k},$$

and

$$\operatorname{Var}(|M|)=(1+o_d(1))\frac{n}{k^2d^{1/k}}.$$

The conjecture extends the graph case and predicts both the local uncovered-vertex probability and the variance of the matching size. The first assertion is the earlier Kahn conjecture and is refuted for every $k\geq 3$ by this paper; the variance assertion is not resolved by the supplied text.