The spectral gap of the 2 - D stochastic Ising model with mixed boundary conditions Preliminary Draft

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l × l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by δl, where δ < 1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on δ), the spectral gap decreases exponentially in l.