The spectral gap of the 2-D stochastic Ising model with nearly single-spin boundary conditions

Abstract

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N-by-N box, with boundary conditions which are ``plus'' except for small regions at the corners which are either free or ``minus.'' The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order \log N. This means that removing as few as O(\log N) plus spins from the corners produces a spectral gap far smaller than the order N^{-2} gap believed to hold under the all-plus boundary condition. Our results are valid at all subcritical temperatures.