Uniform and \(L^ 2\) convergence in one dimensional stochastic Ising models (Q1116561)

We study the rate of convergence to equilibrium of one dimensional stochastic Ising models with finite range interactions. We do not assume that the interactions are ferromagnetic or that the flip rates are attractive. The infinitesimal generators of these processes all have gaps between zero and the rest of their spectra. We prove that if one of these processes is observed by means of local observables, then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap. Moreover this exponential convergence is uniform in the initial configuration.