Glauber dynamics on hyperbolic graphs : boundary conditions and mixing time

We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an $n$-vertex ball of the hyperbolic graph $H(v,s)$, where $v$ is the number of neighbors of each vertex and $s$ is the number of sides of each face, conditioned on having $(+)$-boundary. If $v>4$, $s>3$ and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of $n$. This implies that the mixing time grows at most linearly in $n$, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature $b$. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time