Applications of the stochastic Ising model to the Gibbs states

The stochastic Ising model is used as a tool to prove theorems concerning analyticity of the correlation functions and strong cluster properties of the Gibbs states.Research supported in part by N.S.F. Grant MPS74-18926.     

A simple PDE and Wiener‐Hopf Riccati equations

We investigate the PDE (1.1), concentrating on the case when σ < in which the boundary condition (1.1b) is not of Feller's type and we lose the minimum principle. Investigation of nonnegative solutions leads us to Wiener‐Hopf theory and to a Riccati equation. A much‐studied Markov chain analogue is developed further in the hope that it will illuminate all aspects of the PDE case. © 2005 Wiley Periodicals, Inc.     

Uniform andL 2 convergence in one dimensional stochastic Ising models

We study the rate of convergence to equilibrium of one dimensional stochastic Ising models with finite range interactions. We donot 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.The authors were partially supported by N.S.F. Grants DMS-8609944 and DMS-8611487, respectively. Also, the second author acknowledges supports from DAAL 03-86-K-0171     

Logarithmic Sobolev inequalities and stochastic Ising models

We use logarithmic Sobolev inequalities to study the ergodic properties of stochastic Ising models both in terms of large deviations and in terms of convergence in distribution.