  We study the stochastic Ising model on finite graphs with n vertices and
bounded degree and analyze the effect of boundary conditions on the mixing
time. We show that for all low enough temperatures, the spectral gap of the
dynamics with (+)-boundary condition on a class of nonamenable graphs, is
strictly positive uniformly in n. This implies that the mixing time grows at
most linearly in n. The class of graphs we consider includes hyperbolic graphs
with sufficiently high degree, where the best upper bound on the mixing time of
the free boundary dynamics is polynomial in n, with exponent growing with the
inverse temperature. In addition, we construct a graph in this class, for which
the mixing time in the free boundary case is exponentially large in n. This
provides a first example where the mixing time jumps from exponential to linear
in n while passing from free to (+)-boundary condition. These results extend
the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable
graphs.
