Conference Agenda

We have proved the multidimensional version of a Poisson process approximation theorem for specific point processes. This result can be used to derive multidimensional compound Poisson limit theorems in the total variation distance for vectors whose components are all $U$-statistics of the same Poisson point process. In general, the components have to be asymptotically independent, but we will also show how one can proceed if this is not the case. Our examples are based on the Gilbert graph.

The generalised random graph contains $n$ vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. In the following, we require certain moment assumptions concerning the weights, ranging from finite second to finite fourth moments. The object of interest is the point process $\mathcal{C}_n$ on $\{3,4,…\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson process. When $\mathcal{C}_n$ is evaluated on a bounded set $A$, we provide a rate of convergence in the total variation distance. If the graph is subcritical, $A$ is allowed to be unbounded, which comes at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and of the longest cycle when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung-Lu model and the Norros-Reittu model.

A process of a random planar quadrangulation is introduced as an approximation for certain cellular automata describing neuronal dynamics. This model turns out to be a particular (rectangular) case of Gilbert tessellation. The initial state of the model is a Poisson point process on a plane with intensity $\lambda$. Each point is assigned independently equally probable direction, horizontal or vertical. With time, the rays grow from each point with a constant speed into both sides along the given direction. As soon as a ray crosses the pathway of another ray, it stops growing.

The central and still open question is the distribution of the line segments. We derive exponential bounds for the tail of this distribution. The correlations in the model are derived for some instructive examples. In particular, the exponential decay of correlations is established. For the initial conditions confined in a box $[0,N]^2\subset\mathbb{R}^2$ it is proved that the average number of rays escaping the box has a linear order in $N$.

Let $\eta$ be an isometry invariant Poisson process of $k$-flats, $0\leq k\leq d-1$, in $d$-dimensional hyperbolic space.

For $d-m(d-k)\geq 0$, the $m$-th order intersection process of $\eta$ consists of all nonempty intersections of distinct flats $E_1,\ldots,E_m\in\eta$.

Of particular interest is the total volume $F^{(m)}_r$ of this intersection process in a ball of radius $r$, since it has been observed that this functional obeys a nonstandard CLT for certain values of $d,k$.

For $2k>d+1$, we determine the asymptotic distribution of $F^{(m)}_r$, as $r\to\infty$, previously known only for $m=1$, and derive rates of convergence in the Kolmogorov distance. We further discuss properties of the non-Gaussian limit distribution.