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Rigorous Analysis of Low-Temperature Stochastic Ising Models: Metastability and Exponential Approach to Equilibrium

We present rigorous results on the exponential convergence to equilibrium for the Metropolis and Swendsen-Wang stochastic dynamics for the 2d-Ising ferromagnet in the low-temperature case with external magnetic field h in the thermodynamic limit. We consider various situations in which boundary conditions are homogeneous and parallel or opposite to the external field. In the latter case for the SW dynamics we also analyse the tunnelling from the metastable phase to the stable one.