  We reconsider the problem of local persistence in directed site percolation.
We present improved estimates of the persistence exponent in all dimensions
from 1+1 to 7+1, obtained by new algorithms and by improved implementations of
existing ones. We verify the strong corrections to scaling for 2+1 and 3+1
dimensions found in previous analyses, but we show that scaling is much better
satisfied for very large and very small dimensions. For d > 4 (d is the spatial
dimension), the persistence exponent depends non-trivially on d, in qualitative
agreement with the non-universal values calculated recently by Fuchs {\it et
al.} (J. Stat. Mech.: Theor. Exp. P04015 (2008)). These results are mainly
based on efficient simulations of clusters evolving under the time reversed
dynamics with a permanently active site and a particular survival condition
discussed in Fuchs {\it et al.}. These simulations suggest also a new critical
exponent $\zeta$ which describes the growth of these clusters conditioned on
survival, and which turns out to be the same as the exponent, \eta+\delta in
standard notation, of surviving clusters under the standard DP evolution.
