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

TL;DR

This paper proves a Mermin--Wagner type theorem for Gibbs states on infinite random Lorentzian triangulations, showing the absence of spontaneous continuous symmetry-breaking in models of quantum gravity with classical spins.

Contribution

It establishes a Mermin--Wagner theorem for Gibbs states on Lorentzian triangulations related to critical Galton--Watson trees, extending symmetry results to quantum gravity models.

Findings

Gibbs measures are ${\tt G}$-invariant on almost all Lorentzian triangulations.

No spontaneous continuous symmetry-breaking occurs in the studied models.

The theorem applies to a broad class of random triangulations linked to critical Galton--Watson trees.

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