Probabilidade. Teoria e Aplicações
Modelos unidimensionais com interação de longo alcance: percolação e transição de fase
Pretendo fazer uma revisão geral do tema e discutir alguns resultados recentes, baseados principalmente em trabalhos em colaboração com M. Cassandro, I. Merola, D. Marchetti, V. Sidoravicius e S. Friedli.
$ell_1$ estimation of unbounded pairwise interaction of a Gibbs measure
"In Neuroscience, to infer how neurons interact to each other is an important problem to understand how brain works. Currently, there is no technique that makes possible to infer the connectivity of more than hundreds of neurons. As a possible solution to this, we propose as a model of interaction between neurons the Gibbs measure on $mathbb{Z}^d$ having long range interaction and, as the estimation procedure, the $ell_1$ regularized pseudo-maximum likelihood.
Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata
Using an extended version of the duality concept between two stochastic processes, we give ergodicity conditions for two states probabilistic cellular automata (PCA) of any dimensions and any radius. Under these assumptions, in the one dimensional case, we study some properties of the unique invariant measure and show that it is shift mixing. Also, the decay of correlation is studied in detail.
