  We study continuous time Glauber dynamics for random configurations with
local constraints (e.g. proper coloring, Ising and Potts models) on finite
graphs with $n$ vertices and of bounded degree. We show that the relaxation
time
  (defined as the reciprocal of the spectral gap $|\lambda_1-\lambda_2|$) for
the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$.
For these hyperbolic graphs, this yields a general polynomial sampling
algorithm for random configurations. We then show that if the relaxation time
$\tau_2$ satisfies $\tau_2=O(1)$, then the correlation coefficient, and the
mutual information, between any local function (which depends only on the
configuration in a fixed window) and the boundary conditions, decays
exponentially in the distance between the window and the boundary. For the
Ising model on a regular tree, this condition is sharp.
