A Mermin-Wagner Theorem for Gibbs States on Lorentzian Triangulations

M. Kelbert, Yu Suhov, A. Yambartsev

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Abstract

We establish a Mermin-Wagner type theorem for Gibbs states on infinite random Lorentzian triangulations (LT) arising in models of quantum gravity. Such a triangulation is naturally related to the distribution P of a critical Galton-Watson tree, conditional upon non-extinction. At the vertices of the triangles we place classical spins taking values in a torus M of dimension d, with a given group action of a torus G of dimension d′≤d. In the main body of the paper we assume that the spins interact via a two-body nearest-neighbor potential U(x,y) invariant under the action of G. We analyze quenched Gibbs measures generated by U and prove that, for P-almost all Lorentzian triangulations, every such Gibbs measure is G-invariant, which means the absence of spontaneous continuous symmetry-breaking.

Original languageEnglish (US)
Pages (from-to)671-677
Number of pages7
JournalJournal of Statistical Physics
Volume150
Issue number4
DOIs
Publication statusPublished - Feb 11 2013

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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