  This is the second of a series of two papers dealing with local limit
theorems in relatively hyperbolic groups. In this second paper, we restrict our
attention to non-spectrally degenerate random walks and we prove precise
asymptotics of the probability $p_n(e, e)$ of going back to the origin at time
$n$. We combine techniques adapted from thermodynamic formalism with the rough
estimates of the Green function given by the first paper to show that $p_n(e,
e) \sim CR^{-n} n^{-3/2}$ , where $R$ is the spectral radius of the random
walk. This generalizes results of W. Woess for free products and results of
Gou{\"e}zel for hyperbolic groups.
