The Asymmetric One-Dimensional Constrained Ising Model: Rigorous Results

We study a one-dimensional spin (interacting particle) system, with product Bernoulli(p) stationary distribution, in which a site can flip only when its left neighbor is in state +1. Such models have been studied in physics as simple exemplars of systems exhibiting slow relaxation. In our “East” model the natural conjecture is that the relaxation time τ(p), that is 1/(spectral gap), satisfies log τ(p) ∼ log (1/p) log 2 as p ↓ 0. We prove this up to a factor of 2. The upper bound uses the Poincaré comparison argument applied to a “wave” (long-range) comparison process, which we analyze by probabilistic techniques. Such comparison arguments go back to Holley (1984, 1985). The lower bound, which atypically is not easy, involves construction and analysis of a certain “coalescing random jumps” process.