Filters

2015 - 20192

More filters

2019 - 201912020 - 20221
Reset filters

Settings

Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Interacting particle systems ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Attractor properties for irreversible and reversible interacting particle systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Jahnel, Benedikt; Külske, Christof
    We 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.