Conference Agenda

The finite-dimensional Schatten-$p$ classes $S_p^{m \times n}$, consisting of real or complex $m \times n$-matrices endowed with the $\ell_p$-norm of their singular values, are a classical object of study in functional analysis and also have proved their usefulness in applications (e.g. compressed sensing). In addition to their functional analytic qualities, the geometric properties of the corresponding unit balls are subject to intense ongoing research, both in the exact and the asymptotic regimes. Yet most results so far have been obtained for the square case, i.e. $m = n$.

In this talk we will present recent findings about the unit balls of Schatten-$p$ classes of not necessarily square matrices. Among those are the exact volume of the $S_\infty^{m \times n}$-unit ball, a Poincaré-Maxwell-Borel principle for the uniform distribution on the $S_\infty^{m \times n}$-unit ball, and a Sanov-type large deviations principle for the empirical measure of the singular values of a random matrix sampled uniformly from the $S_p^{m \times n}$-unit ball, for any $0 < p \leq \infty$.

Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in \mathbb{N}$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$; we also consider geometric quantities other than the volume. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture, in particular they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719].

I will present the paper with the same title as this talk. In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying Poisson point process $\eta$. The main result are new concentration bounds of the form $$\mathbb{P}(|F_m ( f , \eta) -\mathbb{E} F_m ( f , \eta)| \ge t)\leq 2\exp(-I(\gamma,t)),$$ where $I(\gamma,t)$ is of optimal order in $t$, namely it satisfies $I(\gamma,t)=\Theta(t^{1\over m}\log t)$ as $t\to\infty$ and $\gamma$ is fixed. The function $I(\gamma,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\Lambda$. One of the key ingredients of the proof is bounding the centred moments of $F_m(f,\eta)$. We discuss the optimality of obtained concentration bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.

This is a joint work with Anna Gusakova.

Gibbs processes in the continuum are one of the most fundamental models in spatial stochastics. They are typically defined using a density with respect to the Poisson point process. In the language of statistical mechanics, this corresponds to the grand-canonical ensemble, where the number of particles is random. Of the same importance is the canonical ensemble, where the number of particles is fixed. In the language of point processes, this corresponds to studying binomial Gibbs processes which are defined using a density with respect to the binomial point process.

In this talk, we present a large deviation theory developed for functionals of binomial Gibbs processes with fixed intensity in increasing windows. Our method relies on the traditional large deviation result from [1] noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle delicate and unlikely pathological events. The presented results cover a broad class of both the interaction function (possibly unbounded) and the functionals (given as a sum of possibly unbounded local score functions).

[1] Georgii, H.-O. and Zessin, H. (1993): Large deviations and the maximum entropy principle for marked point random fields, Probab. Theory Related Fields 96, 177-204.