Convergence in L2 of Stochastic Ising Models: Jump Processes and Diffusions

Abstract

This chapter investigates the convergence of the semigroups of stochastic Ising models in the L 2 spaces of their Gibbs (equilibrium) states. The chapter finds conditions that guarantee that the convergence takes place exponentially fast. Exponentially fast L 2 convergence has several interesting consequences. The system returns to equilibrium quickly if it is displaced from equilibrium by a small amount. It gives quite a bit of information on the space time correlations when the system is in equilibrium. This information can be used to study renormalization and the resulting limiting process. If the interaction is attractive (ferromagnetic) then it is possible to draw conclusions about the rate of convergence in the uniform norm from information about the rate of convergence in L 2 . The chapter explains the consequences of exponentially fast convergence in L 2 . The chapter focuses on stochastic Ising models in which changes take place via continuous diffusion with reflection at the ends of a bounded interval, and chooses the diffusion and drift coefficients to minimize the technical difficulties in the construction of these processes. The chapter describes these processes and the corresponding Gibbs states.