The asymmetric one-dimensional constrained Ising model

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)∼\(\tfrac{{\log ^2 (1/p)}}{{\log 2}}\) as p↓0. 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.

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