Probability and Ergodic Theory

Researchers

Karine Bertin, Sebastián Donoso, Joaquín FontbonaRaúl Gouet, Rodolfo Gutiérrez, Alejandro MaassServet Martínez, Daniel RemenikJaime San Martín, Avelio Sepúlveda, Soledad Torres

About the research group

This group puts together two research lines with a long-shared history within CMM, based on common interests and activities. The group works on a variety of fundamental topics in probability theory, stochastic processes, dynamical systems and ergodic theory, and in applications to other branches of pure and applied mathematics such as number theory, geometry, mathematical physics and mathematical biology. At the same time, members of the group develop theory and applications in a diverse range of subjects where randomness and information theory play a key role, including data science, systems biology and bioinformatics, astroinformatics, mining and geophysics.

An important focus of work is stochastic models in mathematical physics. A highlight of the group’s research in this direction has been KPZ fixed point, the universal Markov process describing the asymptotic spatial fluctuations of all models in the KPZ universality class, and its surprising connections with dispersive PDEs. Another focus of interest is the long time behavior of interacting particle systems, where the group has made important contributions to the development of probabilistic techniques for propagation of chaos in mean field models in kinetic theory, such Kaç particle systems associated with the Boltzmann equation. A more recent focus of interest is two-dimensional random geometry. Here members of the group have made key contributions to the study of the Gaussian free field, which has been key to further understand properties of fundamental models in two-dimensional statistical physics including random curves such as the Schramm-Loewner evolution and random metric spaces such as Liouville quantum gravity; one highlight is the discovery of a topological phase transition for the Gaussian free field.

Quasi-stationary distributions and M-matrices are two areas of long-standing interest for the group, which have led to two monographs describing the state of the art of these theories. Other work by the group focuses on stochastic modeling in genetics and population dynamics, on finite potential theory, and on stochastic analysis, particularly the characterization of multi-dimensional local martingales and its relation to the phenomenon of bubbles in finance. A recent focus of interest is the application of ideas from optimal transport in data science.

In ergodic theory and dynamical systems, contributions were made in several directions, with connections to number theory, combinatorics, analysis, group theory and geometry. Deep results were obtained in Teichmüller dynamics, such as the resolution of the famous Kontsevich-Zorich conjecture on the simplicity of the Lyapunov spectrum for quadratic differentials. In addition, a program in symbolic dynamics was developed attacking fundamental problems for minimal Cantor systems, establishing a correspondence between finite rank systems and S-adic subshifts, and providing new tools to tackle several problems related to the complexity of minimal subshifts.

Other important results deal with recurrence problems both in ergodic theory and topological dynamics, such as the analysis of multicorrelation sequences for Zd actions and the study of sets of recurrence for actions of the multiplicative semigroup of integers.