Simons Workshop on Stochastic Geometry and Point Processes

 

Talk Abstracts (Click to show/hide)

Random Spatial Networks

Aldous, D., UC Berkeley, USA

The talk will give an overview of my interests in this topic, emphasizing:
  • There are models for connected networks other than the familiar ``geometric random graph" model and its variants.
  • Believing one can describe complex real-world networks by highly specific models with only few parameters seems unduly optimistic. As an alternative, one could consider some initially unspecified model within a specified large class. (analogy: natural language modeled as some initially unspecified stationary process). For instance, the property that route-lengths are linear should hold very generally.
  • There is a (presumably) large class of scale-invariant models with mathematically interesting structure.

Clustering comparison of point processes with applications to percolation

Blaszczyszyn, B., Ecole Normale Superieure, France

We review some recent results involving comparison of clustering properties of point processes. The whole approach is funded on some basic observations allowing to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving the void probabilities and moment measures, thus allowing to study the impact of clustering of the underlying point process. When stronger tools are needed, dcx ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is concerned. We will sketch some recent results obtained using the aforementioned comparison tools, in particular regarding percolation of the Boolean Model. Based on joint work with D. Yogeshwaran.

Construction of fractal random series with point processes and Voronoi tessellations

Calka, P., Universite de Rouen, France

Weierstrass and Takagi-Knopp type series are infinite sums of sinusoidal or sawtooth signals with fractal graph. For more than a century, they have been intensively studied as models for irregular signals and rough surfaces. Their definition requires an underlying sequence of partitions of the space which is in general derived from the dyadic cubes. In this talk, we construct a new family of random series defined on $\R^d$. They are close to the Takagi-Knopp model with the extra ingredient of a sequence of random partitions of the space, distributed as Voronoi tessellations of an i.i.d. sequence of underlying point processes. These series are continuous functions, indexed by one scaling parameter and two Hurst-like exponents. We show that the graph of such function is fractal with explicit and equal box dimension and Hausdorff dimension. The proof of this result relies on the adaptation of general criteria from fractal geometry (oscillation estimates, Frostman-type lemma) on the one hand and new distributional properties of Voronoi tessellations on the other hand. Particular emphasis will be given to the particular ingredients from stochastic geometry. We will finally discuss some extensions of the model. This talk is based on joint works with Yann Demichel.

Yaglom limits via Holley inequality

Ferrari, P., Universidad de Buenos Aires, Argentina

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.

Geometric methods in probability, probabilistic methods in geometry: selected examples

Hug, D., Karlsruhe Institute of Technology, Germany

Geometric methods and ideas abound in Stochastic Geometry. In this talk, we provide some new examples for this interaction of geometry and probability. The first deals with affine diameters of convex bodies. An affine diameter of a convex body is a closed segment connecting two points that lie in distinct parallel supporting hyperplanes of the body. It is known that, in the sense of Baire categories, in {\em most} convex bodies, {\em most} points belong to infinitely many affine diameters. We pursue the somewhat complementary question asking for the {\em average} number of affine normals through the points of a typical convex body. To answer this question in a strengthened form, we prove sharp inequalities for the average number of affine diameters through the points of a convex body $K\subset\R^n$ in the cases where $K$ is a polytope, $K$ is sufficiently smooth and for strictly convex $K$ if $n=2$. In a second example, we consider stationary random processes of $k$-dimen\-sio\-nal affine subspaces (flats) in $\R^n$. If $2k\geq n$, then intersection processes are investigated, while in the complementary case $2k < n$ a proximity process is introduced. The intensity measures of these processes are described in terms of parameters of the underlying $k$-flat process. By a translation into geometric parameters of associated zonoids and by means of integral transformations, several new uniqueness and stability results for the underlying process of $k$-flats are derived. They rely on a combination of known and novel estimates for area measures of zonoids. An asymptotic second-order analysis as well as central and non-central limit theorems for length-power direction functionals of proximity processes complement earlier works for intersection processes. The first part of the talk is based on joint work with Imre B\'ar\'any and Rolf Schneider, the second part concerns joint work with Christoph Th\"ale and Wolfgang Weil.

Normal approximation of geometric Poisson functionals

Last, G., Karlsruhe Institute of Technology, Germany

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.

Shape Matters: Physics of Stochastic Geometries

Mecke, K., Universitat Erlangen-Nurnberg, Germany

Spatially structured matter such as foams, gels or biomaterials are of increasing technological importance due to their shape-dependent material properties. But the shape of disordered structures is a remarkably incoherent concept and cannot be captured by correlation functions alone. Integral geometry furnishes a suitable family of morphological descriptors, so-called tensorial Minkowski functionals, which are related to curvature integrals and do not only characterize shape but also anisotropy and even topology of disordered structures. These measures can be used to derive structure-property relations for complex materials.

Stabilization via semigroup interpolations

Peccati, G., Luxemburg University, Luxemburg

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 Poincaré inequalities and a variant of Mehler's formula. Joint work with G. Last and M. Schulte (Karlsruhe).

Rigidity and tolerance for perturbed lattices

Peres, Y., Microsoft Research

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).

Multipodal Phases in Dense Networks

Radin, C. UT Austin, USA

We consider asymptotically large, dense, simple graphs constrained in both their edge and triangle densities, and study the associated entropy, the goal being to determine the structure of `exponentially most' graphs with given variable constraints. We have proven there is a sharp phase transition between graphs of large versus low edge density E, at any fixed triangle density T, when T=E^3. Simulations show that the high edge density regime is further broken up into well-defined regions which are overwhelmingly dominated by graphs of very simple `multipodal' structure, simple modifications of balanced multipartite graphs. This is joint work with Kui Ren and Lorenzo Sadun.

Shift-Coupling and Mass-Stationarity

Thorisson, H., University of Iceland

Variance asymptotics and scaling limits for Gaussian polytopes

Yukich, J., Lehigh University, USA

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.