Variance asymptotics and scaling limits for Gaussian polytopes (J. Yukich)

Consider the convex hull of n i.i.d. random variables with the standard normal distribution on d-dimensional Euclidean space. As n tends to infinity, we establish variance asymptotics for the volume and k-face functionals of the convex hull, resolving an open problem. Asymptotic variances and the scaling limit of the boundary of the convex hull are given in terms of functionals of germ-grain models having parabolic grains with apices at a thinned non-homogeneous Poisson point process. The scaling limit of the boundary of the convex hull coincides with that featuring in the geometric construction of the zero-viscosity solution of Burgers’ equation. This is joint work with P. Calka.

Stabilization via semigroup interpolations (G. Peccati)

I will present some recent advances in the domain of limit theorems for geometric Poisson functionals. The main result is a general (optimal) Berry-Esseen bound for stabilizing functionals, based on iterated PoincarĂ© inequalities and a variant of Mehler’s formula. Joint work with G. Last and M. Schulte (Karlsruhe).

Minicourse: Invariant Matching (Y. Peres)

Suppose that red and blue points occur as independent point processes in R^d, and consider translation-invariant schemes for perfectly matching the red points to the blue points. (Translation-invariance means that the matching is constructed in a way that does not favor one spatial location over another). What is best possible cost of such a matching, measured in terms of the edge lengths? What happens if we insist that the matching is non-randomized, or if we forbid edge crossings, or if the points act as selfish agents? I will discuss recent progress and open problems on these topics, as well as on the related topic of fair allocation. In particular I will address some new discoveries on multi-color matching and multi-edge matching.

Yaglom Limits via Holley Inequality (P. Ferrari)

We consider Markov chains on a countable partially ordered state space with an absorbing state. Assume that the absorbed chain has a quasi stationary distribution. We give sufficient conditions on the transition probabilities to guarantee that the law of the chain at time n conditioned to non absorption is monotone non decreasing in n. As a consequence the Yaglom limit (the limit of the conditioned chain starting from a minimal state, as n goes to infinity) converges to a quasi stationary distribution; the limit is minimal for the stochastic order of measues. The approach uses a dynamics on the space of trajectories and Holley inequality. Joint work with Leonardo Rolla.

Rigidity and tolerance for perturbed lattices

Consider a perturbed lattice {v+Y_v} obtained by adding IID d-dimensional Gaussian variables {Y_v} to the lattice points in Z^d. Suppose that one point, say Y_0, is removed from this perturbed lattice; is it possible for an observer, who sees just the remaining points, to detect that a point is missing? Holroyd and Soo (2011) noted that in one and two dimensions, the answer is positive: the two point processes (before and after Y_0 is removed) can be distinguished using smooth statistics, analogously to work of Sodin and Tsirelson (2004) on zeros of Gaussian analytic functions. The situation in higher dimensions is more delicate, with a phase transition that depends on a game-theoretic idea, in one direction, and on the unpredictable paths constructed by Benjamini, Pemantle and the speaker (1998), in the other. I will also describe a related point process where removal of one point can be detected but not the removal of two points. (Joint work with Allan Sly, UC Berkeley).

Normal approximation of geometric Poisson functionals (G. Last)

We consider functionals of a (possibly marked) stationary spatial Poisson processes. Recent progress on Stein’s method shows how to use stochastic analysis on Poisson spaces for bounding the Wasserstein and the Kolmogorov distance between the distribution of a Poisson functional (suitably normalized) and the standard normal distribution in terms of Malliavin operators. In the first part of this talk we show that this method can be successfully applied to the intrinsic volumes (and more general additive functionals) of the Boolean model and of intersection processes of Poisson hyperplanes. In the second part we will apply a second order Poincare inequality involving only first and second order differential operators to the Poisson-Voronoi tessellation and nearest neighbour graphs. This talk is based on joint work with Daniel Hug, Giovanni Peccati, Mathew Penrose Matthias Schulte and Christoph Thaele.

More videos

The complete list of talk and mini courses videos of the 2015 conference on stochastic geometry and point processes can be found here.


Simons Conference on Networks and Stochastic Geometry Videos, May 18-21, 2015