  We study speeds of fronts in bistable, spatially inhomogeneous media at
parameter regimes where speeds approach zero. We provide a set of conceptual
assumptions under which we can prove power-law asymptotics for the speed, with
exponent depending a local dimension of the ergodic measure near extremal
values. We also show that our conceptual assumptions are satisfied in a context
of weak inhomogeneity of the medium and almost balanced kinetics, and compare
asymptotics with numerical simulations.
