  We consider an Ising ferromagnet endowed with zero-temperature spin-flip
dynamics and examine the evolution of the Ising quadrant, namely the spin
configuration when the minority phase initially occupies a quadrant while the
majority phase occupies three remaining quadrants. The two phases are then
always separated by a single interface which generically recedes into the
minority phase in a self-similar diffusive manner. The area of the invaded
region grows (on average) linearly with time and exhibits non-trivial
fluctuations. We map the interface separating the two phases onto the
one-dimensional symmetric simple exclusion process and utilize this isomorphism
to compute basic cumulants of the area. First, we determine the variance via an
exact microscopic analysis (the Bethe ansatz). Then we turn to a continuum
treatment by recasting the underlying exclusion process into the framework of
the macroscopic fluctuation theory. This provides a systematic way of analyzing
the statistics of the invaded area and allows us to determine the asymptotic
behaviors of the first four cumulants of the area.
