"Geometrical optics" in Brownian motion
One of the simplest dynamical models in statistical mechanics is Brownian motion, which describes the random movement of a particle that diffuses in space. What is the likelihood for a Brownian particle to travel very far in a short space of time, given some constraints and/or external forces? To answer such questions, we use a formalism that is very similar to geometrical optics: We calculate the most likely trajectory of the particle conditioned on the constraints, by solving an optimisation problem. This formalism is the main idea that we used in the papers 1 and 2 and we have many ideas for using it in additional systems.
Anomalous scalings of large deviations of time averages:
In a broad class of classical dynamical systems (called ergodic systems), the average value of a physical observable over a long period of time converges to its average over the system's steady-state distribution, meaning that time averages become equivalent to ensemble averages. In large deviation theory, however, one is often interested in fluctuations of such time averages, that is, in the probability that they will be significantly different from their expected value. This could be of interest, for instance, in order to predict the likelihood of unusually high or low temperatures or rainfall, averaged over long periods of time. In two recent works, 1 and 2, we studied such questions in two different standard theoretical models in statistical physics (Ornstein-Uhlebleck and fractional Brownian motion with stochastic resetting). We found anomalous scalings, which cause large deviations in these models to be far likelier than one would naively expect.
Counting statistics for trapped fermions and random matrix theory
One way to investigate many-body quantum effects is to study cold atoms: gases of particles cooled down to very low temperatures. I am interested in spatial properties of such gases, in particular of fermions that are trapped by an external potential. By a mathematical miracle, the positions of the fermions are in some cases related to eigenvalues of random matrices. In the papers 1, 2 and 3, we studied counting statistics: fluctuations of the number of fermions in a specified domain in space (and, analogously, fluctuations of the number of eigenvalues of a random matrix that belong to some subset), and related questions.