TITLE: Concentration Bound of $0/1$ permanent
QUESTION [7 upvotes]: If I pick a random $0/1$ $n\times n$ matrix with $0$ occuring with probability $p$ then what does the distribution of the permanent look like?

REPLY [5 votes]: Let $X_n$ denote the permanent of your random 0/1 $n\times n$ matrix.  Then after normalizing, $X_n$ converges to a log-normal distribution [as does the determinant of the matrix].  That is, $\log(|X_n|)$ has a central limit theorem.
For references, see:
https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/div-classtitlethe-numbers-of-spanning-trees-hamilton-cycles-and-perfect-matchings-in-a-random-graphdiv/09FDBF0DCE75F834AEBD831B156753D5
or
https://mini.pw.edu.pl/~wesolo/publikacje/2002RemWesJTP.pdf
Or see the following recent paper that obtains a more nuanced result (as well as concentration results and bounds on moments).
https://arxiv.org/abs/1605.07570