Assume odds of $1$ or $0$ is $0.5$ and independent.
Suppose we have two randomly generated $0$ and $1$ sequences if length $2n$ each and each with number of $1$s between $0.5n - a$ and $0.5n + a$ where $a$ is at most $\sqrt{b n \ln n}$ where $b>0$ is fixed.
What is the probability distribution of number of common $1$s between sequences?
Is there a concentration bound?