MARK KAC SEMINAR

In 2005, Basko, Aleiner and Altschuler claimed that strongly disordered quantum systems exhibit a vanishing heat conductivity, i.e. the diffusion constant is zero. This phenomenon came later to be known under the name "Many-Body-Localization". It is the counterpart of "Anderson localization" for interacting systems and it is probably a purely quantum phenomenon. One can view it as a KAM theorem that remains remarkably valid in the limit of an infinite number of degrees of freedom. Recently, we gave a mathematical proof of this claim, see arXiv:2408.04338. This is joint work with Lydia Giacomin, Francois Huveneers, and Oskar Prosniak. I will try to sketch the setup and the mathematical challenges.

There are several functional encodings of random trees which are commonly used to prove (among other things) scaling limit results. We consider two of these, the height process and Lukasiewicz path, in the classical setting of a branching process tree with critical offspring distribution of finite variance, conditioned to have n vertices. These processes converge jointly in distribution after rescaling by √n to constant multiples of the same standard Brownian excursion, as n goes to infinity. Their difference (taken with the appropriate constants), however, is a nice example of a discrete snake whose displacements are deterministic given the vertex degrees; to quote Marckert, it may be thought of as a “measure of internal complexity of the tree”. We prove that this discrete snake converges on rescaling by $n^{-1/4}$ to the Brownian snake driven by a Brownian excursion. This is a consequence of our new theory for “globally centred” discrete snakes that improves earlier works of Marckert and Janson and enjoys further applications in, for example, random maps. This is joint work in progress with Louigi Addario-Berry, Christina Goldschmidt and Rivka Mitchell.