Emergence of metastability on the hyperbolic lattice: Effects of boundary conditions

Abstract

We investigate the Ising model on finite subgraphs of the hyperbolic lattice under minus boundary conditions and in the presence of a positive external field $h$. Interpreting the boundary as frozen or cold wall conditions, we show that, for small values of $h$, the system exhibits metastable behaviour. Our result is very surprising, since non-amenable graphs, such as hyperbolic lattices, feature exponentially growing boundaries, which typically destabilize local energy minima. In particular, we identify the unique metastable state and characterize the exit time from it. Furthermore, we establish asymptotic results for the distribution of the first hitting time and provide estimates for the spectral gap. Finally, we analyze the energy landscape and describe the nucleation mechanism for values of $h$ outside the metastable regime.