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Author: Jahnel, Benedikt × End date: 2023 × Start date: 2020 × Has files: Yes × WGL Institute: WIAS ×

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Now showing 1 - 10 of 24
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    Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures
    ([Madralin] : EMIS ELibEMS, 2018) Cotar, Codina; Jahnel, Benedikt; Külske, Christof
    The 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.
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    Absence of percolation in graphs based on stationary point processes with degrees bounded by two
    (New York, NY [u.a.] : Wiley, 2022) Jahnel, Benedikt; Tóbiás, András
    We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for signal-to-interference ratio graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional k-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.
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    Phase transitions for chase-escape models on Poisson–Gilbert graphs
    ([Madralin] : EMIS ELibEMS, 2020) Hinsen, Alexander; Jahnel, Benedikt; Cali, Elie; Wary, Jean-Philippe
    We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The 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.
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    Exponential Moments for Planar Tessellations
    (New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Jahnel, Benedikt; Tóbiás, András
    In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl 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.
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    Sharp phase transition for Cox percolation
    (Seattle, Wash. : Univ. of Washington, Mathematics Dep., 2022) Hirsch, Christian; Jahnel, Benedikt; Muirhead, Stephen
    We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.
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    Lower large deviations for geometric functionals
    ([Madralin] : EMIS ELibEMS, 2020) Hirsch, Christian; Jahnel, Benedikt; Tóbiás, András
    This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson–Voronoi cells, as well as power-weighted edge lengths in the random geometric, k-nearest neighbor and relative neighborhood graph.
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    Percolation and connection times in multi-scale dynamic networks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Jahnel, Benedikt; Hirsch, Christian; Cali, Eli
    We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.
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    Phase 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-Philippe
    We 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.
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    Continuum percolation in a nonstabilizing environment
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Jhawar, Sanjoy Kumar; Vu, Anh Duc
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].
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    Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Jahnel, Benedikt; Külske, Christof
    We consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [0,1]. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large p, as it changes only a small fraction p(1-p)2d of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.