S(P)AM annual colloquium - June 2, 2026.

Macroscopic Fluctuations far from equilibrium, large deviations and integrability. Many natural systems continuously exchange matter, energy or information with their surroundings. Due to these permanent fluxes that break time-reversal invariance, such systems are far from thermodynamic equilibrium and lie beyond the traditional principles of equilibrium statistical mechanics. For instance, the steady state of a conducting rod in contact with two reservoirs at different temperatures can not be accounted for by any of the classical ensembles of statistical mechanics. The quest for a unified approach to non-equilibrium behaviour, that would go far beyond linear response theory, has remained an open problem for many years. In the last decades, the discoveries of certain universal laws and exact solutions of paradigmatic models have unveiled some of the mysteries of non-equilibrium physics. In parallel, large deviation theory has emerged as an adequate framework to formulate general properties like the fluctuation relations, and variational principles such as the macroscopic fluctuation theory of Gianni Jona-Lasinio and his collaborators. The goal of this talk is to present some of these concepts and to explain how a quantitative understanding of non-equilibrium fluctuations at different levels (microscopic, mesoscopic and macroscopic) has be achieved with the help of various integrability techniques, inspired from quantum physics and from inverse scattering theory.

Random graphs as models of quantum disorder. A disordered quantum system is mathematically described by a large Hermitian random matrix. One of the most remarkable phenomena expected to occur in such systems is a localization-delocalization transition for the eigenvectors. Originally proposed in the 1950s to model conduction in semiconductors with random impurities, the phenomenon is now recognized as a general feature of wave transport in disordered media, and is one of the most influential ideas in modern condensed matter physics. A simple and natural model of such a system is given by the adjacency matrix of a random graph. I report on recent progress in analysing the phase diagram for the Erdös-Renyi model of random graphs. In particular, I explain the emergence of fully localized and fully delocalized phases, which are separated by a mobility edge. I also explain how to obtain optimal delocalization bounds using a new Bernoulli flow method. Based on joint work with Johannes Alt, Raphael Ducatez, and Joscha Henheik.