  We revisit the problem of local persistence in directed percolation,
reporting improved estimates of the persistence exponent in 1+1 dimensions,
discovering strong corrections to scaling in higher dimensions, and
investigating the mean field limit. Moreover, we introduce a graded persistence
probability that a site does not flip more than n times and demonstrate how
local persistence can be studied in seed simulations. Finally, the problem of
spatial (as opposed to temporal) persistence is investigated.
