Distribution of $k$-matchings in a random graph

Take the Erdos-Renyi random graph $$G(n,p)$$, i.e. the random graph with $$n$$ vertices and where each possible edge has an independent probability of $$p$$ of being present. Recall that a $$k$$-matching is a disjoint union of $$k$$ edges. Let $$X$$ be a random variable equal to the number of subgraphs in $$G(n,p)$$ which are isomorphic to the $$k$$-matchings. I want to investigate the distribution of $$X$$.

Clearly for any fixed set of $$2k$$ vertices $$(x_1, y_1, x_2, y_2, ..., x_k,y_k)$$ from $$G(n,p)$$ the probability that there is an edge from ever $$x_i$$ to every $$y_i$$ is $$p^{k}$$. Therefore $$\mathbb{E}(X) = (2k-1)!!\binom{n}{2k}p^{k}$$. Note here double factorial !! is the product of all even numbers up to $$(2k-1)$$ of the same parity. But what is the (approximate) distribution of $$X$$.

For concreteness, I am interested in situations where say $$p= \frac{1}{3}$$ and I'd like to bound the tail probabilities `Chernoff style', i.e. I'd like to be able to bound from above $$\mathbb{P}(X \le \frac{1}{a} \mathbb{E}(X)\,)$$. Again for concretness say set $$a=5$$.

I'm sure this must be a studied problem but I'm not familiar with the literature in this area so any pointers would be greatly appreciated!

Edit: It turns out that from Theorem 5.6. here https://www.math.cmu.edu/~af1p/BOOK.pdf we know the distribution is normal. All that remains is to work out the variance of $$X$$. In fact there is a later section on tail probabilities which is also useful.

• Why is your expectation correct? You should be choosing $2k$ vertices, not just $k$. Can you revise your calculations? Commented Jul 6, 2023 at 23:57
• Also, why $p^{-k}$ and not $p^k$? Commented Jul 7, 2023 at 0:23
• thank you I've edited my question Commented Jul 7, 2023 at 7:42
• @InuyashaYagami apologies, this factor was not so important to me as it only depends on $k$ but I should have been more careful before posting. Commented Jul 7, 2023 at 8:13