METASTABILITY AND SPINODAL POINTS FOR A RANDOM WALKER ON A TRIANGLE

We investigate time-dependent properties of a single-particle model in which a random walker moves on a triangle and is subjected to nonfoca l 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 brea king) allows us to study the spectrum of the time evolution operator. We break the symmetry explicitly and find a stable phase, and a metast able phase which vanishes at a spinodal point. At this point, the spec trum of the time evolution operator has a gapless and universal band o f excitations with a dynamical critical exponent z = 1. Surprisingly, the imaginary parts of the eigenvalues E-j(L) are equally spaced, foll owing the rule JE(j)(L) proportional to j/L. Away from the spinodal po int, we find two time scales in the spectrum. These results are relate d to scaling functions for the mean path of the random walker and to f irst passage times. For the spinodal point, we find universal scaling behavior. A simplified version of the model which can be handled analy tically is also presented.