  We simulate the formation of cosmic strings at the zeros of a complex
Gaussian field with a power spectrum $P(k) \propto k^n$, specifically
addressing the issue of the fraction of length in infinite strings. We make two
improvements over previous simulations: we include a non-zero random background
field in our box to simulate the effect of long-wavelength modes, and we
examine the effects of smoothing the field on small scales. The inclusion of
the background field significantly reduces the fraction of length in infinite
strings for $n < -2$. Our results are consistent with the possibility that
infinite strings disappear at some $n = n_c$ in the range $-3 \le n_c < -2.2$,
although we cannot rule out $n_c = -3$, in which case infinite strings would
disappear only at the point where the mean string density goes to zero. We
present an analytic argument which suggests the latter case. Smoothing on small
scales eliminates closed loops on the order of the lattice cell size and leads
to a ``lattice-free" estimate of the infinite string fraction. As expected,
this fraction depends on the type of window function used for smoothing.
