Cycles of given lengths in unicyclic components in sparse random graphs

Let $L$ be subset of $\{3,4,\dots \}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1\mathrm{/}2$ and the critical regime $M=\frac{n}{2}(1+\mu n^{-1\mathrm{/}3})$ with $\mu =O(1)$. Depending on the regime and a condition involving the series ${\sum }_{l\in L}\frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to \mathrm{\infty }$.