  The presence of random fields is well known to destroy ferromagnetic order in
Ising systems in two dimensions. When the system is placed in a sufficiently
strong external field, however, the size of clusters of like spins diverges.
There is evidence that this percolation transition is in the universality class
of standard site percolation. It has been claimed that, for small disorder, a
similar percolation phenomenon also occurs in zero external field. Using exact
algorithms, we study ground states of large samples and find little evidence
for a transition at zero external field. Nevertheless, for sufficiently small
random field strengths, there is an extended region of the phase diagram, where
finite samples are indistinguishable from a critical percolating system. In
this regime we examine ground-state domain walls, finding strong evidence that
they are conformally invariant and satisfy Schramm-Loewner evolution
($SLE_{\kappa}$) with parameter $\kappa = 6$. These results add support to the
hope that at least some aspects of systems with quenched disorder might be
ultimately studied with the techniques of SLE and conformal field theory.
