 
Summary: Journal of Statistical Physics. Vol. 92, Nos. 5/6. 1998
Metastability and Spinodal Points for a Random
Walker on a Triangle
Peter F. Arndt1 and Thomas Heinzel1
Received January 27, 1997: final June 3, 1998
We investigate timedependent properties of a singleparticle model in which
a random walker moves on a triangle and is subjected to nonlocal boundary
conditions. This model exhibits spontaneous breaking of a Z2 symmetry. The
reduced size of the configuration space (compared to related manyparticle
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 metastable 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 Ej(L) are equally spaced,
following the rule JEj(L) oc 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.
