The threshold for fractional clique decompositions of random hypergraphs, via matrix scaling

Let ${\delta }_{k,r}^{\ast }$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge 2$, $r\ge k+1$ and $\epsilon >0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $({\delta }_{k,r}^{\ast }+\epsilon )n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p\ge C(\log n\mathrm{/}{n}^{r-k}{)}^{1\mathrm{/}(\left(\genfrac{}{}{0px}{}{r}{k}\right)-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.