Colloque - Géométrie et spectre des grands objets - Alice Guionnet : About Non-Commutative Entropy and Topology

Nalini AnantharamanGéométrie spectraleCollège de FranceAnnée 2025-2026Colloque - Géométrie et spectre des grands objets - Alice Guionnet : About Non-Commutative Entropy and TopologyAlice GuionnetCNRS, École normale supérieure de LyonIn the 1990s, Voiculescu developed the theory of non-commutative entropy. For a single non-commutative variable, this entropy reduces to the rate function of the empirical measure of the eigenvalues of a Gaussian matrix. For several non-commutative variables, such a principle of large deviations, concerning the joint moments of Gaussian matrices, is not completely established. The topology used is that of the weak topology of non-commutative laws. This topology is not adequate for studying matrices whose coefficients have heavy tails and which typically have a finite number of non-zero coefficients per row or column. To overcome this shortcoming, Camille Male introduced traffics and the corresponding topology. I will discuss the associated entropy introduced in a recent article with Charles Bordenave and Camille Male, as well as the relationship between the topology of traffics and Benjamini and Schramm topology.