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A Mermin–Wagner Theorem for Gibbs States on Lorentzian Triangulations

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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.

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Acknowledgements

This work was supported by FAPESP 2012/04372-7. M.K. thanks FAPESP 2011/20133-0 and thanks NUMEC for kind hospitality. The work of A.Y. was partially supported CNPq 308510/2010-0.

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Swansea, UK

    M. Kelbert

  2. Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, UK

    Yu. Suhov

  3. IITP, Russian AS, Moscow, Russia

    Yu. Suhov

  4. Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil

    M. Kelbert, Yu. Suhov & A. Yambartsev

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  1. M. Kelbert
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  2. Yu. Suhov
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  3. A. Yambartsev
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Correspondence to A. Yambartsev.

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Kelbert, M., Suhov, Y. & Yambartsev, A. A Mermin–Wagner Theorem for Gibbs States on Lorentzian Triangulations. J Stat Phys 150, 671–677 (2013). https://doi.org/10.1007/s10955-013-0698-8

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  • DOI: https://doi.org/10.1007/s10955-013-0698-8

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