MARK KAC SEMINAR

The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract GREM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes thetrue cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. Joint works with Nicola Kistler, and Marius A. Schmidt.

As seen in the previous lectures, Derrida's Random Energy Models have played a key role in the understanding of certain issues in spin glasses. The mathematical analysis of these models - in particular the multi-scale refinement of the second moment method as devised by Kistler, is also particularly efficient to analyse the so-called log-correlated class; the latter consists of Gaussian fields with - as the name suggests, logarithmically decaying correlations. I will introduce/recall some models falling into this class, and the main steps in their analysis through the paradigmatic Branching Brownian motion / Branching Random Walk. Finally, I will conclude with recent results on models which are not even Gaussian, but for which the multiscale treatment still goes through swiftly: the directed and undirected first passage percolation in the limit of large dimensions, a.k.a. the (un)directed polymers in random environment. Joint works with Nicola Kistler, and Marius A. Schmidt.

Internal DLA models the growth of a random discrete set by subsequent aggregation of particles. At each step, a new particle starts inside the current aggregate, and it performs a simple random walk until reaching an unoccupied site, where it settles. The large scale properties of IDLA clusters are by now well understood. In these two talks I will instead focus on Internal DLA on cylinder graphs, seen as a Markov chain on the space of particle configurations. I will present several techniques for bounding the maximal fluctuations of IDLA clusters, which allow one to show that the stationary distribution concentrates on a small subset of the infinite state space. I will then discuss the mixing time of the chain, and its dependence on the choice of the cylinder's base graph.