## Preprints

• On maximal hard-core thinnings of stationary particle processes arXiv
joint with G. Last
The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally unique. More precisely, distributional uniqueness holds in subcritical and barely supercritical regimes of continuum percolation. Additionally, based on the analysis of a specific example, we argue that fluctuations in grain sizes can play an important role for establishing distributional uniqueness at high intensities. Finally, we provide a family of algorithmically constructible approximations whose volume fractions are arbitrarily close to the maximum.
• Uniformity of hitting times of the contact process arXiv
joint with M. Heydenreich and D. Valesin
For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times $t(x)$, defined for each site $x$ as the first time at which it becomes infected. First, the family of random variables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, is stochastically tight. Second, for each $\varepsilon >0$ there exists $x$ such that, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probability larger than $1-\varepsilon$. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchand (Ann. Appl. Probab., 2012).

## Publications in Conference Proceedings

• R. Shah, C. Hirsch, D.P. Kroese and V. Schmidt, Rare event probability estimation for connectivity of large random graphs. Proceedings of the 2014 Winter Simulation Conference, A. Tolks, S.D. Diallo, I.O. Ryzhov, L. Yilmaz, S. Buckley, and J.A. Miller, eds
• D. Neuhaeuser, C. Hirsch, C. Gloaguen and V. Schmidt, A parametric copula approach for modelling shortest-path trees in telecommunication networks. In: A. Dudin and K. Turck (eds.) Analytical and Stochastic Modeling Techniques and Applications. Lecture Notes in Computer Science 7984, Springer, Berlin 2013, 324-336.