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- ItemExtremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures([Madralin] : EMIS ELibEMS, 2018) Cotar, Codina; Jahnel, Benedikt; Külske, ChristofThe concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.
- ItemPhase transitions for chase-escape models on Gilbert graphs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hinsen, Alexander; Jahnel, Benedikt; Cali, Eli; Wary, Jean-PhilippeWe present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
- ItemAttractor properties for irreversible and reversible interacting particle systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Jahnel, Benedikt; Külske, ChristofWe consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.
- ItemDisruptive events in high-density cellular networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Keeler, Paul; Jahnel, Benedikt; Maye, Oliver; Aschenbach, Daniel; Brzozowski, MarcinStochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signal-to-interference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the near-far effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.
- ItemSharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Jahnel, Benedikt; Külske, ChristofWe investigate the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments
- ItemThe typical cell in anisotropic tessellations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Hirsch, Christian; Jahnel, Benedikt; Hinsen, Alexander; Cali, ElieThe typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.
- ItemSINR percolation for Cox point processes with random powers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Jahnel, Benedikt; Tóbiás, AndrásSignal-to-interference plus noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.
- ItemExponential moments for planar tessellations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Tóbiás, András; Jahnel, BenediktIn this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.
- ItemPercolation for D2D networks on street systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Cali, Elie; En-Najjari, Taoufik; Gafur, Nila Novita; Hirsch, Christian; Jahnel, Benedikt; Patterson, Robert I.A.We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.
- ItemLarge deviations for the capacity in dynamic spatial relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hirsch, Christian; Jahnel, BenediktWe derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.