Spring day in probability
University of Florence
Friday 26 May 2017
Lecturers: Robert Morris (IMPA) and Roberto Fernandez (Utrecht)
Location: Aula Magna di Via S. Gallo 10, Firenze
Prof. Robert Morris
Title: Monotone cellular automata
Abstract: Cellular automata are interacting particle systems whose update rules are local and homogeneous. Since their introduction by von Neumann almost 50 years ago, many particular such systems have been investigated, but no general theory has been developed for their study, and for many simple examples surprisingly little is known. Understanding their (typical) global behaviour is an important and challenging problem in statistical physics, probability theory and combinatorics.
In this talk I will outline some recent progress in understanding the behaviour of a particular (large) family of monotone cellular automata -- those which can naturally be embedded in d-dimensional space -- with random initial conditions. For example, in the case where a site updates (from inactive to active) if at least r of its neighbours are already active, these models are known as emph{bootstrap percolation}, and have been extensively studied for various specific underlying graphs. Apart from their inherent mathematical interest, the study of these processes is motivated by their close connection to models in statistical physics, and I will discuss some applications to a family of models of the liquid-glass transition known as emph{kinetically constrained spin models}.
Prof. Roberto Fernandez
Title: Signal description: process or Gibbs?
Abstract: The distribution of signals such as spike trains is naturally modeled through stochastic processes where the probability of future states depend on the pattern of past spikes. Mathematically, this corresponds to distributions *conditioned on the past*. From a signal-theoretic point of view, however, one could wonder whether a more efficient description could be obtained through the simultaneous conditioning of past *and* future. Furthermore, such a formalism could be appropriate when discussing string without a particular "time" order, such as the distribution of DNA nucleotides, or even issues related to anticipation and prediction in neuroscience. On the mathematical level this double conditioning would correspond to a Gibbsian description analogous to the one adopted in statistical mechanics. In this talk I will introduce and contrast both approaches ---process and Gibbsian based--- reviewing existing results on scope and limitations of them.
Program:
11.00-11.45 Introductory lecture: Fernandez
11.45-12.00 Break
12.00-12.45 Seminar: Fernandez
13.00-14.30 Lunch
14.30-15.15 Introductory lecture: Morris
15.15-15.30 Break
15.30-16.15 Seminar: Morris