Normality conjecture for cycle and matching counts in random regular graphs

Let ${\mathcal G}(n,d)$ be the uniform random $d$-regular graph on $n$ vertices. For an integer $\ell$, an $\ell$-cycle is a cycle with $\ell$ vertices, and an $\ell$-matching is a matching with $\ell$ edges.

Normality conjecture. Suppose

$$\min\{d,n-d\}\to\infty$$

as $n\to\infty$. The number of $\ell$-cycles in ${\mathcal G}(n,d)$ is asymptotically normal for all $3\le \ell\le n$, and the number of $\ell$-matchings in ${\mathcal G}(n,d)$ is asymptotically normal for all $3\le \ell\le n/2$.

This conjecture asserts that normal limiting behavior persists across the full ranges of cycle and matching sizes, extending the paper's discussion of possible distributional phase transitions for these counts.