  We present very accurate numerical estimates of the time and size dependence
of the zero-temperature local persistence in the $2d$ ferromagnetic Ising
model. We show that the effective exponent decays algebraically to an
asymptotic value $\theta$ that depends upon the initial condition. More
precisely, we find that $\theta$ takes one universal value $0.199(2)$ for
initial conditions with short-range spatial correlations as in a paramagnetic
state, and the value $0.033(1)$ for initial conditions with the long-range
spatial correlations of the critical Ising state. We checked universality by
working with a square and a triangular lattice, and by imposing free and
periodic boundary conditions. We found that the effective exponent suffers from
stronger finite size effects in the former case.
