The volume of simplices in high-dimensional Poisson–Delaunay tessellations

Gusakova, Anna; Thäle, Christoph

Annales Henri Lebesgue, Tome 4 (2021), pp. 121-153.

Nous considérons des simplexes aléatoires pondérés typiques $Z_{\mu }$, $\mu \in (-2,\infty )$, dans une tessellation de Poisson–Delaunay dans ${ℝ}^n$, où le poids est donné par la puissance $(\mu +1)$ du volume. Ceci inclut en particulier les simplexes de Poisson–Delaunay dans les cas typiques ($\mu =-1$) et pondérés par le volume ($\mu =0$). En prouvant des bornes précises sur les cumulants, nous montrons que le volume logarithmique de $Z_{\mu }$ satisfait un théorème central limite en grande dimension, i.e., lorsque $n\to \infty$. De plus, nous établissons des vitesses de convergence. Parallèlement, nous étudions les inégalités de concentration et les déviations modérées. Ce cadre permet au poids $\mu =\mu \left(n\right)$ de dépendre également de la dimension. Nous discutons séparément un certain nombre de cas particuliers. Lorsque $\mu$ est fixé, nous étudions également la convergence mod-$\varphi$ et les grandes déviations du volume logarithmique de $Z_{\mu }$.

Typical weighted random simplices $Z_{\mu }$, $\mu \in (-2,\infty )$, in a Poisson–Delaunay tessellation in ${ℝ}^n$ are considered, where the weight is given by the $(\mu +1)$st power of the volume. As special cases this includes the typical ($\mu =-1$) and the usual volume-weighted ($\mu =0$) Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_{\mu }$ satisfies a central limit theorem in high dimensions, that is, as $n\to \infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $\mu =\mu \left(n\right)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\varphi$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{\mu }$ are investigated.

Classification : 52A22, 60D05, 60F05, 60F10
Mots clés : Berry–Esseen bound, central limit theorem, cumulants, high dimensions, mod-$\phi $ convergence, moderate deviations, large deviations, random simplex, Poisson–Delaunay tessellation, stochastic geometry

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