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 $c<1/2$ and the critical regime $M=frac{n}{2}left(1+mu n^{-1/3}right)$ 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 $ntoinfty$.