We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study the distributional limit of their stationary distributions when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. Several examples are discussed.

Disordered systems such as glasses and spin glasses pose a challenge to theoretical physics. As a simplification mean-field models where the geometry of interactions is induced by a random graph have been introduced. These models turn out to be intimately related to deep questions in combinatorics that have been studied in their own right for well over half a century. Some of these questions have exciting applications in statistics and computer science. In this talk I am going to give an overview of this interdisciplinary research area and of the new contributions that physics insights have enabled.

Consider Bessel, Jacobi, and Dunkl processes on RN which depend on several positive coupling constants k . These processes describe interacting particle systems of Calogero-Moser-Sutherland type with N particles. For some cases, these processes are related with β -Hermite, -Laguerre, and -Jacobi ensembles. Moreover, for the frozen case k=∞ , the processes degenerate to deterministic or pure jump processes. We show how the generators for these processes lead to analogues of Wigner's semicircle and Marchenko-Pastur limit laws for N→∞ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In the frozen cases, our approach leads to a new proof of well known limit laws for the empirical measures of the zeroes of Hermite, Laguerre, and Jacobi polynomials. The talk is based on joint work with J. Woerner and M. Auer.

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

The assumption that high dimensional data is Gaussian is pervasive in many statistical procedures, due to the ease of analytic tractability this special distribution provides. We explore the relaxation of the Gaussian assumption in Shrinkage estimation, Stein's Unbiased Risk Estimate, and Single Index Models, using two tools that originate in Stein's method: Stein kernels, and the zero bias distribution. Taking this approach leads to measures of discrepancy from the Gaussian that arise naturally from the form of the procedures considered, and result in performance bounds that apply in contexts not restricted to the Gaussian. The resulting bounds typically include an additional term that reflects the cost of deviation from the Gaussian, and that vanishes for the Gaussian, thus recovering this paricular special case. (with Xiaohan Wei, Max Fathi, Gesine Reinert, and Adrien Samaurd)

In this talk, we discuss some limitations of the quantum graph approach in defining Dirichlet forms, Laplacians and/or Brownian motions in case when the graph has vanishing edge lengths. Our model set is the stretched Sierpinski gasket, SSG for short. It is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. (It has also been called ``Hanoi attractor``.) As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing together the Brownian motions on the intervals. In fact, there have been several works in this direction. However, there still remains, ``reminiscence`` of the Sierpinski gasket in the geometric as well as analytic structure of SSG and the same should therefore be expected for diffusions. This is a joint work with Patricia Alonso Ruiz (Texas A&M University) and Jun Kigami (Kyoto University).