Buongiorno a tutti,
segnalo il seguente seminario:
*Lunedì 4 dicembre, ore 17:00*
Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
*Speaker:* Daniel Parisi, Sapienza Università di Roma
*Title:* Entropy and mixing time of non-local Markov chains.
*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