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- 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.
- ItemSpace-time large deviations in capacity-constrained relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hirsch, Christian; Jahnel, Benedikt; Patterson, RobertWe consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.
- ItemGibbsian representation for point processes via hyperedge potentials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Jahnel, Benedikt; Külske, ChristofWe consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.
- ItemGibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Jahnel, Benedikt; Külske, ChristofWe 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.
- ItemThe Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Jahnel, Benedikt; Külske, ChristofWe consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG > 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > tG, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.
- ItemDynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Köppl, JonasWe consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.
- ItemExtremal decomposition for random Gibbs measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.
- ItemLower large deviations for geometric functionals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hirsch, Christian; Jahnel, Benedikt; Tóbiás, AndrásThis 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, κ-nearest neighbor and relative neighborhood graph.
- ItemGibbsianness of locally thinned random fields(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Engler, Nils; Jahnel, Benedikt; Külske, ChristofWe consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions
- ItemContinuum percolation in a nonstabilizing environment(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Jhawar, Sanjoy Kumar; Vu, Anh DucWe 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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