Rapid convergence to equilibrium in one dimensional stochastic Ising models. (English) Zbl 0558.60077

One considers the stochastic Ising model in one dimension, which is a Markov process on \(\{-1,1\}^ Z\) with semigroup \(T_ t=\exp (t\cdot L)\), where \[ Lf(\eta)=\sum_{k\in Z}c_ k(\eta)\cdot (f(\eta^ k)- f(\eta)). \] (\(\eta\) \({}^ k\) is obtained from \(\eta\) by reversing the spin \(\eta_ k\) at site k.) The flip rates \(c_ k\) are supposed to be strictly positive, compatible with translations and depending only on finitely many coordinates. Furthermore, the process satisfies the detailed balance (reversibility) condition with respect to the Gibbs state \(\mu\) of a certain finite range potential. The result says that a certain universality in the ergodic behavior of the process holds, independently of the details of the dynamics given by the \(c_ k:\) one has always exponential decay to equilibrium in the following sense: Theorem 0.4. There is a \(\gamma >0\) such that for each \(f\in L^ 2(\mu)\) with \(\int fd\mu =0\) one has in \(L^ 2(\mu)\) \[ \| T_ Tf\|_ 2\leq e^{-\gamma t}\cdot \| f\|_ 2,\quad t>0. \] Theorem 0.6. If in addition the rates \(c_ k\) are attractive, i.e. if \((T_ t)\) preserves stochastic order, then there is a constant \(\delta >0\) such that for each cylindrical function f with \(\int fd\mu =0\) one has \[ \sup_{\eta}| T_ tf(\eta)| \leq A\cdot e^{-\delta t}\quad for\quad all\quad t>0, \] with A depending on f.
Reviewer: H.Rost


60K35 Interacting random processes; statistical mechanics type models; percolation theory
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