  We investigate time-dependent properties of a single particle model in which
a random walker moves on a triangle and is subjected to non-local boundary
conditions. This model exhibits spontaneous breaking of a Z_2 symmetry. The
reduced size of the configuration space (compared to related many-particle
models that also show spontaneous symmetry breaking) allows us to study the
spectrum of the time-evolution operator. We break the symmetry explicitly and
find a stable phase, and a meta-stable phase which vanishes at a spinodal
point. At this point, the spectrum of the time evolution operator has a gapless
and universal band of excitations with a dynamical critical exponent z=1.
Surprisingly, the imaginary parts of the eigenvalues E_j(L) are equally spaced,
following the rule Im E_j(L)\propto j/L. Away from the spinodal point, we find
two time scales in the spectrum. These results are related to scaling functions
for the mean path of the random walker and to first passage times. For the
spinodal point, we find universal scaling behavior. A simplified version of the
model which can be handled analytically is also presented.
