  Consider an irreducible Markov chain which satisfies a ratio limit theorem,
and let $\rho$ be the spectral radius of the chain. We investigate the relation
of the the $\rho\,$-Martin boundary with the boundary induced by the
$\rho\,$-harmonic kernel which appears in the ratio limit. Special emphasis is
on random walks on non-amenable groups, specifically, free groups and
hyperbolic groups.
