The loop erased exit path and the metastability of a biased vote process

The reduction method provides an algorithm to compute large deviation estimates of (possibly nonreversible) Markov processes with exponential transition rates. It replaces the original graph minimisation equations of Freidlin and Wentzell by more tractable path minimisation problems. When applied to study the metastability of the dynamics, it gives a large deviation principle for the loop erased exit path from the metastable state. To illustrate this, we study a biased majority vote process generalising the one introduced in Chen (1997. J. Statist. Phys. 86 (3/4), 779–802). We show that this nonreversible dynamics has a two well potential with a unique metastable state, we give an upper bound for its relaxation time, and show that for small enough values of the bias the exit path is typically different at low temperature from the typical exit paths of the Ising model.