It is in the natural attitude of the human being to find regularities in the midst of chaos. Whether these regularities really underlie the physical nature of the world, or are just hypnotic patterns that satisfy our compelling
need of order, has been a philosophical question since the dawn of our species.
Most of the time, the patterns we see are not that well defined. It is like looking at them through frosted lens glasses: patterns appear blurry, and incomplete. If one looks at them only once they may think that what they see is not even a pattern. Though, if one looks again, and again, and again, it starts to become clear that there is some consistency and uniformity in what is seen. In particular, if all these different 'pictures' are superimposed, the result seems indeed to evoke a clear figure.
I am attracted by comprehending how these blurry motives, or fluctuations, arise in out-of-equilibrium systems, i.e., irreversible systems characterised by non-zero fluxes. In particular, I am interested in understanding how likely are these fluctuations to appear and what are the physical mechanisms that generate them.
Many of the tools and mathematical techniques I use are inherited from the theory of large deviations: a branch of probability theory that deals with sums of random variables which decay exponential fast in the size of the system, e.g. time in the study of nonequilibrium.
I list here topics, and project, I like and work on.
A rigorous treatment of solvable models of equilibrium or nonequilibrium systems is faced here. The interest is in developing mathematical methods that can be applied to study more general systems and in particular
Fluctuations arise in (finite size) random processes. Here, by means of large deviation theory, one can estimate their likelihood to appear.
Another interesting question is to understand what are the physical processes that generate these fluctuations. Once this is clear, one could develop tools to control the appearance of fluctuations.
These are processes that have the property of being re-initialised at random times to a specific initial condition. The queue at the front-office, or the motion of a protein in a cell are only two examples of the wide range
of applications that reset processes have in modeling real-world scenarios.
Here the focus is again on studying their fluctuations by means of large deviation theory. Recently, we worked on ratio observables, showing that their large deviations have universal features, as for instance differentiability, and tails bounded from above. In particular we noticed the lack of a maximum, which is instead present in the large deviations of the stochastic efficiency (also a ratio). This opens new research horizons...
Here the interest is in the study of systems composed by many interacting components, and the disorder is usually encoded in having different two-body interactions.
Many examples of disordered systems, e.g., ecosystems and plasmas, are carried as applications of random matrix theory. In particular, we focus on the fluctuations of the top eigenvalue of a random matrix. This indeed provides a good playground to study the statistics of strongly correlated random variables.
Random walks on random graphs are not only a good model of disease spreading and information retrieval, but also a good laboratory to study, analytically and numerically, the interplay between two sources of randomness:
the dynamical process itself, and the environment that hosts it.
For instance, a dynamical phase transition in additive observables, e.g. the mean degree visited, or the entropy production of the random walk, appears. As mean field tools are not of great help, we make use of other techniques, such as conditioning on rare events, stochastic adaptive algorithms, and replica calculations, to gain new insights.
These are different dynamical processes, associated here because we make use of similar variational calculation techniques to study them.
Transverse diffusions have the property of converging faster than a normal diffusion to the same invariant measure. For instance, they are useful to accelerate Markov processes used to estimate high-dimensional integrals.
Self-interacting diffusions are processes with long-range interactions in time (memory). This is encoded in the drift of the process, as a dependence on the empirical occupation measure. These are as tough as rich processes to deal with. The goal is to develop a theory of their fluctuations.