The relaxation rate of a stochastic spreading process in a closed ring

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.