# Workshop: High dimensional spatial random systems

Dates: February 22-26, 2021

Venue: Online

Organizers: Apostolos Giannopoulos (Athen), Friedrich Götze (Bielefeld), Matthias Reitzner (Osnabrück), Christoph Thäle (Bochum)

Description: concentration phenomena, high dimensional limit theorems, probabilistic facets of high dimensional convex bodies, random convex bodies, random graphs, random polytopes, random simplicial complexes, random tessellations

Speakers:

Francois Baccelli, Erik Broman, Günter Last, Ilya Molchanov, Tobias Müller, Eliza O’Reilly, Peter Pivovarov, Ngoc Mai Tran, D. Yogeshwaran

In addition, three minicourses will be held:

Omer Bobrowski: Random simplicial complexes
Zakhar Kabluchko: Random polytopes
Joscha Prochno: The large deviations approach to high-dimensional convex bodies

Talks are given online and access data will be sent to the registered participants in due course. Recordings of the talks will be posted online.

## Video recordings and slides

Omer Bobrowski: Random Simplicial Complexes

A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicial complexes, as an extension of random graph theory. In addition to the interesting new mathematical theory that arises, these structures are also highly useful in modern applications of data and network analysis. In this minicourse we aim to provide an introduction to this field.

Slides Lecture 1

Slides Lecture 2

Slides Lecture 3

Lecture I

Lecture II

Lecture III

Zakhar Kabluchko: Random Polytopes

In these three lectures we will provide an introduction to the subject of beta polytopes. These are random polytopes defined as convex hulls of i.i.d. samples from the beta density proportional to $(1 − ∥x∥^2)^{β}$ on the d-dimensional unit ball. Similarly, beta’ polytopes are defined as convex hulls from the beta’ density proportional to $(1 + ∥x∥^{2})^{−β}$ on the d-dimensional Euclidean space. We shall review various models of stochastic geometry which can be reduced to the beta and beta’ polytopes. These include random cones in a half-space, the Poisson zero cell, the typical Poisson-Voronoi cell, and the corresponding objects on the sphere. We will show how various functionals of these models can be expressed through the expected internal and external angles of beta and beta’ simplices and, if time allows, show how these angles can be computed explicitly. The talks are based on the following papers: arXiv 1801.08008, 1805.01338, 1911.07221, 1907.07534, 1901.10528, 1905.01533, 1909.13335.

Slides (all lectures, without the paintings)

Lecture I

Lecture II

Lecture III

Joscha Prochno: The large deviations approach to high-dimensional convex bodies

Given any isotropic convex body in high dimension, it is known that its typical random projections will be approximately standard Gaussian. The universality in this central limit perspective restricts the information that can be retrieved from the lower-dimensional projections. In contrast, the speeds and rate functions in large deviation principles (LDPs), which describe the fluctuations beyond the normal scale, are known to be non-universal and distribution-dependent. In this sense, the large deviation behavior of a random projection of a convex body depends on the geometry of the underlying convex body or, in other words, LDPs allow one to distinguish high-dimensional probability measures via their lower-dimensional projections. This line of research was initiated by Gantert, Kim, and Ramanan in 2017 and quite a number of results have been obtained in the past three years. In this series of lectures we shall start with an introduction to the basic concepts of large deviations theory. We then move on to discuss some of the contributions that have appeared and present a recent result of Kim, Liao, and Ramanan in which LDPs are obtained under an asymptotic thin shell condition, complementing the central limit theorem for convex sets.

Slides (all lectures)

Lecture I

Lecture II

Lecture III

Ngoc Mai Tran: Stochastic geometry to generalize the Mondrian process

The Mondrian process is a stochastic process that produces a recursive partition of space with random axis-aligned cuts. Random forests and Laplace kernel approximations built from the Mondrian process have led to efficient online learning methods and Bayesian optimization. By viewing the Mondrian process as a special case of the stable under iterated tessellation (STIT) process, we utilize tools from stochastic geometry to resolve some fundamental questions concerning the Mondrian process in machine learning. This talk outlines our main results and layout the key questions at the novel intersection of stochastic geometry and machine learning. Joint work with Eliza O’Reilly.

Francois Baccelli: High dimensional stochastic geometry in the Shannon regime

This talk will focus on Euclidean stochastic geometry in the Shannon regime. In this regime, the dimension n of the Euclidean space tends to infinity, point processes have intensities which are exponential functions of n, and the random compact of interest sets have diameters of order square root of n. Three basic models of stochastic geometry based on Poisson processes will be considered: the Boolean model, the Poisson-Voronoi tessellation and the Poisson hyperplane tessellation. We will show how to calculate the asymptotic behavior of classical quantities of stochastic geometry for these three models in the Shannon regime: for the Boolean model, the volume fraction, the percolation threshold or the size of the connected components; for tessellations, statistics of geometric properties of cells (volume, diameter, etc.). These questions are motivated by problems in information theory. This includes the calculation of the Shannon capacity and that of error exponents for channel coding based on random codes and the evaluation of distortion in one-bit-compression based source coding. This talk is a survey of work in collaboration with Venkat Anantharam (UC Berkeley) and Eliza O’Reilly (Caltech).

Slides

Eliza O’Reilly: Facets of high dimensional random polytopes

We consider the model of n i.i.d. points chosen uniformly from the unit sphere in R^d and study the asymptotic behavior of the (d−1)-dimensional faces, or facets, of the convex hull of these points. In fixed dimension d, known asymptotic formulas as the number of points n grows provide results on approximation of the sphere and random spherical Delaunay tessellations. In joint work with Gilles Bonnet, we generalize these results to the case where both the number of points n and the space dimension d are allowed to tend to infinity. The geometry of high dimensional space imposes different regimes for n and d with different asymptotic behavior of the facets. We obtain asymptotic formulas in each case, illuminating the limiting shapes of these polytopes in high dimensions.

Tobias Mueller: Percolation on hyperbolic Poisson-Voronoi tessellations

I will discuss percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. That is, we colour each cell of the hyperbolic Poisson-Voronoi tessellation black with probability p and white with probability 1-p, independently of the colours of all other cells. We say that percolation occurs if there is an infinite connected cluster of black cells. I will sketch joint work with the doctoral candidate Ben Hansen that resolves a conjecture and an open question, posed by Benjamini and Schramm about twenty years ago, on the behaviour of the “critical probability for percolation” as a function of the intensity parameter of the underlying Poisson process. (Unlike in Euclidean Poisson-Voronoi percolation, this critical value depends on the intensity of the Poisson process.) Based on joint work with Benjamin Hansen.

Slides

Erik Broman: Higher dimensional stick percolation

R. Roy introduced the so-called stick percolation model in 1991. This model was a Poisson point process of sticks in $R^2$ where the sticks had random lengths and zero widths. More recently, physicists and chemists have used higher dimensional sticks as a model for studying phenomena such as thin film transistors and conductivity of nano-wires suspended in a non-conductive substance, among other applications. Inspired by these applications and by recent results concerning the so-called Poisson cylinder model, we decided to study the stick model in arbitrary dimensions. In this higher dimensional setting we choose sticks of lengths $L$, widths 1 and of uniform orientation. Of particular interest to us was to investigate how the critical parameter $\lambda_c(L)$ for percolation scales with the length $L$ as $L$ diverges. In this talk I will give a fair amount of background. In addition I will present the main result of an ongoing project which establishes the scaling exponent $\alpha = \alpha (d)$ such that $\lambda_c(L) ∼ L−\alpha$ for every $d \ge 2$.

Guenter Last: Schramm-Steif variance inequalities for Poisson processes and noise sensitivity

Consider a Poisson process η on a general Borel space. Suppose that a square-integrable function f(η) of η is determined by a stopping set Z. Based on the chaos expansion of f(η) we shall de rive analogues of the Schramm-Steif variance inequalities (proved for Boolean functions of independent Rademacher variables). We will show how these inequalities can be used to study quantitative noise sensitivity and exceptional times for binary functions of η. As an application we discuss k-percolation of the Poisson Boolean model with bounded grains. This is joint work with G. Peccati (Luxembourg) and D. Yogeshwaran (Bangalore).

D. Yogeswaran: The Poisson-OSSS inequality and an application to Confetti percolation

I will present a version of the OSSS inequality (proved by O’Donnell, Saks, Schramm and Servedio (2005)) to functionals of general Poisson point processes. This inequality can significantly simplify the proofs of sharp phase-transition in continuum percolation models. We shall illustrate this by an application to confetti percolation model with bounded random grains. Using the sharp phase-transition result, we can also prove that critical probability is 1/2 in certain planar confetti percolation models. A special case of this result was conjectured by Benjamini and Schramm (1998) and proved by Mu ̈ller (2017). Other special cases were proven by Hirsch (2015) and Ghosh and Roy (2018). This is a joint work with Guenter Last (Karlsruhe) and Giovanni Peccati (Luxembourg).

Slides

Peter Pivovarov: Random s-concave functions and isoperimetry

I will discuss stochastic geometry of s-concave functions. In particular, I will explain how a ”local” stochastic isoperimetry underlies several functional inequalities. A new ingredient is a notion of shadow systems for s-concave functions. Based on joint works with J. Rebollo Bueno.

Ilya Molchanov: Random sets generated by translates of a convex body

The K-hull of a compact set A in Euclidean space, where K is a fixed compact convex body, is the intersection of all translates of K that contain A. A set is called K-strongly convex if it coincides with its K-hull. We propose a general approach to the analysis of facial structure of K-strongly convex sets, similar to the well developed theory for polytopes. We then apply our theory in the case when A = Ξn is a sample of n points uniformly distributed on K. We show that in this case the set of x such that x+K contains the sample Ξn, upon multiplying by n, converges in distribution to a zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding f-vector of the K-hull of Ξn to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the f-vector. Joint work with Alexandr Marynych (Kiev).