**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).

- C. Hirsch, B. Jahnel, R.I.A. Patterson, Space-time large deviations in capacity-constrained relay networks (arXiv). Latin American Journal of Probability and Statistics (2017), to appear.
- D. Coupier, C. Hirsch, Coalescence of Euclidean geodesics on the Poisson-Delaunay triangulation (arXiv). Bernoulli (2017), to appear.
- C. Hirsch, B. Jahnel, P. Keeler and R.I.A. Patterson, Large deviations in relay-augmented wireless networks. Queueing Systems (2017), to appear.
- C. Hirsch, B. Jahnel, P. Keeler and R.I.A. Patterson, Traffic flow densities in large transport networks. Advances in Applied Probability 49 (2017), 1091-1115.
- C. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs. Brazilian Journal of Probability and Statistics 31 (2017), 111-143.
- C. Hirsch, B. Jahnel, P. Keeler and R.I.A. Patterson, Large-deviation principles for connectable receivers. Advances in Applied Probability 48 (2016), 1061-1094.
- C. Hirsch, D. Neuhäuser and V. Schmidt, Moderate deviations for shortest-path lengths on random segment processes. ESAIM: Probability and Statistics 20 (2016), 261-292.
- C. Hirsch, Bounded-hop percolation and wireless communication. Journal of Applied Probability 53 (2016), 833-845.
- D. Neuhäuser, C. Hirsch, C. Gloaguen and V. Schmidt, A stochastic model for multi-hierarchical networks. Methodology and Computing in Applied Probability 18 (2016), 1129-1151.
- C. Hirsch, On the absence of percolation in a line-segment based lilypond model. (arXiv). Annales de l’Institut Henri Poincaré 52 (2016), 127-145.
- C. Hirsch, G. W. Delaney and V. Schmidt, Stationary Apollonian packings. Journal of Statistical Physics 161 (2015), 35-72.
- C. Hirsch, D. Neuhäuser, C. Gloaguen and V. Schmidt, Asymptotic properties of Euclidean shortest-path trees in random geometric graphs. Statistics and Probability Letters 107 (2015), 122-130.
- C. Hirsch, G. Gaiselmann and V. Schmidt, Asymptotic properties of collective-rearrangement algorithms. ESAIM: Probability and Statistics 19 (2015), 236-250.
- C. Hirsch, D. Neuhaeuser, C. Gloaguen and V. Schmidt, First-passage percolation on random geometric graphs and an application to shortest-path trees. Advances in Applied Probability 47 (2015), 328-354.
- D. Neuhaeuser, C. Hirsch, C. Gloaguen and V. Schmidt, Joint distributions for total lengths of shortest-path trees in telecommunication networks. Annals of Telecommunications 70 (2015), 221-232.
- C. Hirsch, A Harris-Kesten theorem for confetti percolation. (arXiv). Random Structures & Algorithms 47 (2015), 361-385.
- D. Neuhaeuser, C. Hirsch, C. Gloaguen and V. Schmidt, Parametric modelling of sparse random trees using 3D copulas. Stochastic Models 31 (2015), 226-260.
- T. Brereton, C. Hirsch, V. Schmidt and D. Kroese, A critical exponent for shortest-path scaling in continuum percolation. Journal of Physics A: Mathematical and Theoretical 47 (2014), 505003--505014.
- O. Stenzel, C. Hirsch, V. Schmidt, T. Brereton, D.P. Kroese, B. Baumeier and D. Andrienko, A general framework for consistent estimation of charge transport properties via random walks in random environments. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal 12 (2014), 1108-1134.
- M. C. Christiansen, C. Hirsch and V. Schmidt, Prediction of regionalized car insurance risks based on control variates. Statistics & Risk Modeling 31 (2014), 163-181
- D. Neuhaeuser, C. Hirsch, C. Gloaguen and V. Schmidt, Ratio limits and simulation algorithms for the Palm version of stationary iterated tessellations. Journal of Statistical Computation and Simulation 84 (2014), 1486-1504
- C. Hirsch, D. Neuhaeuser, and V. Schmidt, Connectivity of random geometric graphs related to minimal spanning forests. Advances in Applied Probability 45 (2013), 20-36.
- D. Neuhaeuser, C. Hirsch, C. Gloaguen and V. Schmidt, On the distribution of typical shortest-path lengths in connected random geometric graphs. Queueing Systems 71 (2012), 199-220.

- 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.