  The relaxation process of a diffusive ring becomes under-damped if the bias
(so called affinity) exceeds a critical threshold value, aka delocalization
transition. This is related to the spectral properties of the pertinent
stochastic kernel. We find the dependence of the relaxation rate on the
affinity and on the length of the ring. Additionally we study the implications
of introducing a weak-link into the circuit, and illuminate some subtleties
that arise while taking the continuum limit of the discrete model.
