Probability and Stochastic Processes
Dynamical Phase Transition and Percolation
This project is being carried out by V. Sidoravicius and M. E. Vares. The main international interlocutors are E. Presutti, M. Cassandro, G. Jona Lasinio, F. Martinelli, H. Kesten, C. Newman.
A very active area of research in probability which started to grow since the late 1970's refers to interacting Markovian systems with many components. Basic issues involve the dynamical phase transition, scaling (or hydrodynamical) limits, long time behaviour, metastability, phase separations and related spatial structure. No doubt, this subject offers a great deal of possibilities of interactions with other fields of mathematics and applications to other subjects. The interaction of brazilian probabilists with several research groups in and the US has been extremely fruitful.
Closely related, and coming from honestly applied problem, the mathematical theory of percolation stays for the last two decades with no doubts, as one of the most active and influent subareas of probability theory. Inspite of its applied nature, and sometimes even "easy to state" problems, the questions raised in percolation theory often required to develop new deep theoretical methods and techniques, which lately found broad applications in statistical mechanics, theoretical computer sciences and theoretical probability itself, including here interacting systems.
In the last few years percolation theory faced a new leap due to very fruitful combination of its own "classical" ideas and techniques with methods of complex analysis, ergodic theory and theory of Brownian motion, which led to remarkable progress in scaling theories and conformal invariance. On the other side constantly growing interest from graph-theory and theoretical computer science brought into consideration new important questions and, as a consequence, new ideas and techniques with applications to combinatorial analysis and mathematical logic.
The research V. Sidoravicius and M. E. Vares are developing in this subject is growing in constant collaboration with H. Kesten (Cornell). To exemplify let us quote percolation in a heavily dependent random environment (HDRE), which is described below:
Recently Vares and Sidoravicius observed that some known problems related to the compatibility of binary sequences, which were raised in the late 1980's by information and graph theorists (as for instance P. Winkler, Bell Labs), and which have part of their origins in the theoretical computer sciences, may be reformulated as oriented percolation problems in HDRE. This gives a completely new treatment to a quite complicate combinatorial nature of such questions.
In a sequence of works still in progress (jointly with H. Kesten) we developed a new multiscale renormalization techniques, which we succesfully apply to give an answer to a variety of interesting questions which are either directly related to the compatibility problem, such as contact process with random ranges in a heavily dependent random environment and the traditional oriented percolation in HDRE, or to questions which have independent interest, and in this direction we can rigorously establish a phase transition in percolation of binary sequences on random landscapes.
Currently Vares and Sidoravicius are working on the main question of compatibility of binary sequences. This requires much more elaborated multiscale renormalization techniques combined with new combinatorial estimates.