  We study the thermal equilibrium of nonlinear Klein-Gordon chains at the
limit of small coupling (anticontinuum limit). We show that the persistence
distribution associated to the local energy density is a useful tool to study
the statistical distribution of so-called thermal breathers, mainly when the
equilibrium is characterized by long-lived static excitations; in that case,
the distribution of persistence intervals turns out to be a powerlaw. We
demonstrate also that this generic behaviour has a counterpart in the power
spectra, where the high frequencies domains nicely collapse if properly
rescaled. These results are also compared to non linear Klein-Gordon chains
with a soft nonlinearity, for which the thermal breathers are rather mobile
entities. Finally, we discuss the possibility of a breather-induced anomalous
diffusion law, and show that despite a strong slowing-down of the energy
diffusion, there are numerical evidences for a normal asymptotic diffusion
mechanism, but with exceptionnally small diffusion coefficients.
