The following question seems very natural.

It is a well known consequence of Hall's Theorem that every regular bipartite graph has a perfect matching. Another classical result states that the threshold for a random balanced bipartite graph on $2n$ vertices, in which we keep each vertex with probability $p$, to contain a perfect matching is $p=\frac{ln(n)}{n}$.

The question is, what happens in the intersection of these two cases?

Let $H$ be some $k$-regular balanced bipartite graph on $2n$ vertices, and let $G$ be a random subgraph of $H$ in which we keep each edge with probability $p$, independently of the other edges.

What is the threshold for $G$ to contain a perfect matching? It seems natural to conjecture that when $G$ has no isolated vertices, then with high probability $G$ contains a perfect matching. This happens at $p=\frac{\ln(n)}{k}$.

What I've tried so far is to generalize the proof that the threshold for containing a perfect matching in a random bipartite graph is $p=\frac{\ln(n)}{n}$, which is given here:

https://math.stackexchange.com/questions/1184042/threshold-for-matchings-in-random-bipartite-graphs

The idea is to show that with high probability, the resulting random graph satisfies Hall's condition. The proof relies heavily on the fact that for every set $A$ in the left side and $B$ in the right part, the number of edges between $A$ and $B$ is $|A||B|$. If we knew that our regular graph $H$ satisfied some sort of expander mixing lemma - that is, the number of edges between $A$ and $B$ is always approximately $|A||B|k/n$ - then the proof goes through. However, for a general regular graph $H$ I am still stumped.

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