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 tau(p), that is 1/(spectral gap), satisifies log tau(p) similar to log(2)(1/p)/log 2 as pdown arrow0. We prove this up to a factor of 2. The upper bound uses the Poincare 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.