Georgia Institute of Technology College of Sciences

This is obtained as a limit from the classical Poincar\'e on large random matrices. In the classical case Poincare is obtained in a rather easy way from other functional inequalities as for instance Log-Sobolev and transportation. In the free case, the same story becomes more intricate. This is joint work with Michel Ledoux.

The study of prediction within the realm of Statistical Learning Theory is intertwined with the study of the supremum of an empirical process. The supremum can be analyzed with classical tools:Vapnik-Chervonenkis and scale-sensitive combinatorial dimensions, covering and packing numbers, and Rademacher averages. Consistency of empirical risk minimization is known to be closely related to theuniform Law of Large Numbers for function classes.In contrast to the i.i.d. scenario, in the sequential prediction framework we are faced with an individual sequence of data on which weplace no probabilistic assumptions. The problem of universal prediction of such deterministic sequences has been studied withinStatistics, Information Theory, Game Theory, and Computer Science. However, general tools for analysis have been lacking, and mostresults have been obtained on a case-by-case basis.In this talk, we show that the study of sequential prediction is closely related to the study of the supremum of a certain dyadic martingale process on trees. We develop analogues of the Rademacher complexity, covering numbers and scale-sensitive dimensions, which canbe seen as temporal generalizations of the classical results. The complexities we define also ensure uniform convergence for non-i.i.d. data, extending the Glivenko-Cantelli type results. Analogues of local Rademacher complexities can be employed for obtaining fast rates anddeveloping adaptive procedures. Our understanding of the inherent complexity of sequential prediction is complemented by a recipe that can be used for developing new algorithms.

The usual approach to KPZ is to study the scaling limit of particle systems. In this work, we show that the partition function of directed polymers (with a suitable boundary condition) converges, in a certain regime, to the Cole-Hopf solution of the KPZ equation in equilibrium. Coupled with some bounds on the fluctuations of directed polymers, this approach allows us to recover the cube root fluctuation bounds for KPZ in equilibrium. We also discuss some partial results for more general initial conditions.

We consider random walks on Z^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but which can be arbitrarily close to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point after time 2n. We show that in the situations when the heat kernel exhibits subdiffusive behavior --- which is known to be possible in dimensions d \geq 4-- the walk gets trapped for time of order n in a small spatial region. This proves that the strategy used to infer subdiffusive lower bounds on the heat kernel in earlier studies of this problem is in fact dominant. In addition, we settle a conjecture on the maximal possible subdiffusive decay in four dimensions and prove that anomalous decay is a tail and thus zero-one event. Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.