Gibbs–non-Gibbs properties for evolving Ising models on trees

Aernout C.D. van Enter, Victor N. Ermolaev, Giulio Iacobelli, Christof Külske

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Abstract

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves differently from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad. For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again.
Original languageEnglish
Pages (from-to)774-791
Number of pages18
JournalAnnales de l institut henri poincare-Probabilites et statistiques
Volume48
Issue number3
DOIs
Publication statusPublished - 2012

Keywords

  • Glauber dynamics
  • Cayley tree
  • Tree graphs
  • Ising models
  • Non-Gibbsianness
  • RECOVERY
  • STATE
  • BETHE LATTICE
  • GIBBSIANNESS
  • QUASILOCALITY
  • TRANSITIONS

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