A Mermin--Wagner theorem for Gibbs states on Lorentzian triangulations

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 $\mathsf{P}$ 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 $\mathtt{G}$ of dimension $d^{\prime }\le d$. 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 $\mathtt{G}$. We analyze quenched Gibbs measures generated by $U$ and prove that, for $\mathsf{P}$-almost all Lorentzian triangulations, every such Gibbs measure is $\mathtt{G}$-invariant, which means the absence of spontaneous continuous symmetry-breaking.

10n pages, 1 figure

References in corpus (3)

• The Universe from Scratch
• The spectral dimension of generic trees
• A quantum Mermin--Wagner theorem for quantum rotators on two--dimensional graphs

Cited by in corpus (2)

• A Mermin--Wagner theorem on Lorentzian triangulations with quantum spins
• A quantum Mermin--Wagner theorem for a generalized Hubbard model on a 2D graph