Question 17 of the set of 20 probability problems at https://www.math.ucdavis.edu/~gravner/MAT135A/resources/chpr.pdf provides bounds of $\tfrac{mn}{8}$ and $\tfrac{(m+2)(n+2)}{6}$ for the mean number of orthogonally connected components of an $m$ by $n$ rectangular grid whose squares are randomly (independently and equiprobably) coloured black or white.
Are there any known tighter bounds than this? Or results about any interesting variants of this problem?