Abstract: We discuss the convergence to the stationary distribution for non-local Markov chains on general spin systems on arbitrary graphs. We show that the relative entropy functional of the corresponding Gibbs measure satisfies the block factorization of entropy, an inequality that controls the entropy on a given region V in terms of a weighted sum of the entropies on blocks A ⊂ V when each A is given an arbitrary nonnegative weight α_A. This inequality generalizes the approximate tensorization of entropy and provides a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. As a consequence of block factorization, we obtain optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models, including non-local Markov chains such as the heat-bath block dynamics and the Swendsen-Wang dynamics. The methods also apply to spin systems with hard constraints such as q-colorings and the hard-core gas model. First, we consider spin systems on the d-dimensional lattice Z^d satisfying strong spatial mixing. Then we extend our analysis to spin systems on an arbitrary graph satisfying spectral independence. Finally, we show that the existence of a contractive coupling for any local Markov chain implies spectral independence.

saluti, 

Giacomo Di Gesù