SEMINARIO Speaker: Yinon Spinka Title: Finitely-dependent processes are finitary Venerdi' 18 Gennaio 2019 ORE 11:30 Dipartimento di Matematica e Fisica Universita' degli Studi Roma Tre Largo San Leonardo Murialdo,1 - Pal.C Aula 211 Abstract: Consider a translation-invariant process $X$ indexed by $mathbb{Z}^d$. Suppose that $X$ is finitely-dependent in the sense that its restrictions to sets which are sufficiently separated (at least some fixed distance apart) are independent. We are concerned with the following question: How "close" is $X$ to being an i.i.d. process? One natural notion of closeness, called block factor, was suggested by Ibragimov and Linnik over 50 years ago. It took roughly 30 years until Burton, Goulet and Meester constructed an example which showed that this notion is too strong. That is, $X$ may not be close to being i.i.d. in this sense. We show that $X$ is close in a slightly weaker sense -- it is a finitary factor of an i.i.d. process. This means that $X=F(Y)$ for some i.i.d. process $Y$ and some measurable map F which commutes with translations of $mathbb{Z}^d$, and moreover, that in order to determine the value of $X_v$ for a given $v$, one only needs to look at a finite (but random) region of $Y$. The result extends to finitely-dependent processes indexed by the vertex set of any transitive amenable graph.