The Reduction Method, the Loop Erased Exit Path, and the Metastability of the Biased Majority Vote Process on a Torus

The reduction method provides an algorithm to compute large deviation estimates of (possibly non reversible) 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. We apply this technique to a biased majority vote process generalising the one studied in Chen [6]. We show that this non reversible 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. 1. The model We consider on some finite state space E a family of infinitesimal generators Lβ indexed by a positive real “inverse temperature” parameter β. (Lβf)(σ) = ∑ σ′ 6=σ cβ(σ, σ′)(f(σ′) − f(σ)), where lim β→+∞ − 1 β log cβ(σ, σ′) = V (σ, σ′) ∈ R+ ∪ {+∞}. We let P β t be the semi-group +∞ ∑ n=0 t n! Lβ generated by Lβ and consider on the space D([0,+∞[, E) of right continuous trajectories with left limits the canonical process (Zt)t∈R+ and the family of probability measures P̄σ ∈ M+(D([0,+∞[, E)) of Markov processes starting from σ with semigroups P β t . We know from the matrix tree theorem [13, 11, 7] that the invariant probability distribution μβ of P β t can be expressed with the help of some special families of 1991 Mathematics Subject Classification. 60F10, 60Jxx, 82C05.