### 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 language | English (US) |
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Pages (from-to) | 671-677 |

Number of pages | 7 |

Journal | Journal of Statistical Physics |

Volume | 150 |

Issue number | 4 |

DOIs | |

Publication status | Published - Feb 11 2013 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*150*(4), 671-677. https://doi.org/10.1007/s10955-013-0698-8