The Asymmetric One-Dimensional Constrained Ising Model: Rigorous Results

David Aldous and Persi Diaconis 


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), satisfies
$\log \tau(p) \sim \frac{\log^2 (1/p)}{\log 2} \mbox{ as } p \downarrow 0$.
We prove this up to a factor of $2$.
The upper bound uses the Poincar{\'e} comparison argument applied to a 
``wave" (long-range) comparison process, which we analyze by probabilistic
techniques.  Such comparison arguments go back to Holley (1985).
The lower bound, which atypically is not easy, involves construction and 
analysis of a certain ``coalescing random jumps" process.