SPP2265-Meeting on
Stochastic Geometry
and Point Processes 

In this talk, we consider irreversible translation-invariant interacting particle systems on the d-dimensional hypercubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs with respect to the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure with respect to the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems and is joined work with Jonas Köppl.  

We introduce a non-local transport distance on the space of stationary point processes and analyse the induced geometry. We show - among other things - that the Ornstein-Uhlenbeck semigroup is the gradient flow of the specific relative entropy, 1-geodesic convexity of the specific relative entropy, and functional inequalities such as a Talagrand inequality. Based on joint work with Martin Huesmann.

We study extremal properties of spherical random polytopes, the convex hull of random

points chosen from the unit Euclidean sphere in R^n. The extremal properties of interest

are the expected values of the maximum and minimum surface area among facets.We

determine the asymptotic growth in every fixed dimension, up to absolute constants.

Based on joint work with B. Leroux, L. Rademacher and C. Schuett.

An orthocentric simplex is one whose altitudes intersect in a single point, called the orthocenter. Let $e_0, \ldots, e_d$ be the standard orthonormal basis of $\mathbb{R}^{d+1}$ and let $[...]$ denote the convex hull. Depending on whether the orthocenter of an orthocentric simplex lies in the interior, outside, or on the boundary of the simplex, it is isometric to

$[e_0/\tau_0, e_1/\tau_1, \ldots, e_d/\tau_d]$, $[(\tau_0e_0 +\ldots +\tau_de_d)/(\tau_0^2+\ldots +\tau_d^2), e_1/\tau_1, \ldots, e_d/\tau_d]$, or $[0, e_1/\tau_1, \ldots, e_d/\tau_d]$,

for suitable $\tau_0, \ldots, \tau_d>0$.

We derive explicit formulas for the internal and external angles of those simplices which have an orthocenter in the interior. As a probabilistic application, we obtain explicit expressions for the expected volume of random simplices of the form $[g_1/\tau_1, \ldots, g_n/\tau_n]$, where $g_1, \ldots, g_n$ are independent standard Gaussian distributed vectors in $\mathbb{R}^d$, and $\tau_1, \ldots, \tau_d>0$. The talk is based on joint work with Zakhar Kabluchko.

Topological data analysis (TDA) addresses the problem of distinguishing meaningful topological features from statistical noise. A common tool is the barcode plot, which tracks the appearance and disappearance of topological features as the dataset is examined at varying scales. 

One characteristic of a barcode plot is the inversion count (IC), which counts how many times a bar contains another bar. Another is the tree realization number (TRN), introduced by Scolamiero et al. (2017), which counts the number of equivalence classes of combinatorial trees that can realize a given barcode. 

In this work, we study both the IC and TRN under a geometric null model, using a Poisson point process to generate vertices and constructing commonly used models of trees on top. Our key theoretical contribution is a central limit theorem (CLT) for both IC and TRN. This result enables large-sample approximations that support statistical inference and hypothesis testing in TDA.

This talk is based on ongoing work with Christian Hirsch, Adélie Garin, and Hanna Döring.