## Bernoulli lecture

#### Alice Guionnet will deliver the 2020 Bernoulli - EMS Lecture (nominated by the Bernoulli Society)

## Biosketch

Alice Guionnet is a french mathematician, working in the field of probability theory and more especially statistical mechanics and random matrix theory. She entered Ecole Normale Superieure (Paris) in 1989. She defended her PhD at Université Paris-Sud in 1995 under the supervision of Gerard Ben Arous on Langevin dynamics for spin glasses. She did two short post-doctoral internships at Imperial college and Courant institute (NYU). She has held a position at CNRS since 1993, Berkeley 2005-2006, MIT 2012-2015. She is currently Director of Research (CNRS) at ENS Lyon.

## Research area

Alice Guionnet early works concentrated on large particles systems with random interaction, the so-called spin glasses. These systems model alloys with impurities. She studied their dynamics : because of the randomness of their interactions, these dynamics are not markovian anymore when the number of particles go to infinity (and self-averaging occurs). In particular the long time behaviour of these systems is expected to age, in the sense that their evolution depends on the past on long time scales. Alice Guionnet could characterize with Gerard Ben Arous the asymptotic dynamics of these systems and study their high temperature properties. With Amir Dembo, they could characterize their aging properties for a simpler model called the spherical model.

More recently, she has been studying large random matrices. These mathematical objects appeared in statistics in the work of Wishart to study large arrays of noisy datas. They have since then shown to be ubiquitous to many other fields such as physics or number theory. Together with her collaborators, including Gerard Ben Arous and Ofer Zeitouni, Alice Guionnet investigated the statistical properties of these objects : what can we say about their spectrum when the size of the matrix goes to infinity ? Eventhough Wigner proved in the thirties that if their entries have sufficiently light tail, the empirical measure of these eigenvalues converge to the so-called semi-circle law, many questions were left open : what is happening when the entries have heavy tails ? What can we say about the fluctuations around this limit behaviour ? What is the probability that the spectrum behaves differently and that a large deviation from the typical scheme happens ?

Beyond random matrices, Alice Guionnet was also interested in their connections with free probability and operator algebra. Since the pioneering works of Voiculescu, independent random matrices appeared as useful approximation to free operators and therefore an interesting approach to study them. Alice Guionnet could compare Voiculescu’s entropies in a seminal work with Philippe Biane and Mireille Capitaine. More recently, together with Dima Shlyakhtenko, she could construct free transport, a natural generalization of optimal transport to non-commutative variables. As a by product, they could prove that q-Gaussian C* algebras are isomorphic for sufficiently small q.

Also beyond random matrices, Alice Guionnet studied random particles in strong interactions, as are the eigenvalues of random matrices. She developped with Gaetan Borot a systematic approach to study the fluctuations of such particle systems when they follow the Beta-model distribution. This approach is based on the analysis of an infinite family of equations : the so-called Dyson-Schwinger equations, or loop equations or Master equations.

More recently, Alice Guionnet studied random tilings with Alexei Borodin and Vadim Gorin. Tile the hexagon with lozenges. It turns out there are several possibilities and if one pick a tiling uniformly at random over all these possibilities, the distribution of the lozenges is intimately related with those of the eigenvalues of random matrices, and in fact can be regarded as a discrete analogue of the Beta-models. Using this similarity, and developping the analysis of Nekrasov’s equations, they could study the fluctuations of such random tilings in quite general domains.