Graph dimension and phase transitions

Question

In this question I am mainly concerned with phase transitions associated with symmetry breaking. For phase transitions of this type, there are a number of results describing the relationship between the existence of a phase transition and the dimension of the system.

For example, roughly speaking, the Mermin-Wagner theorem states that there cannot be spontaneous breaking of a continuous symmetry at finite temperature in dimensions $d \leq 2$. This has been proven rigorously for a number of models, including for example the quantum spin-$S$ Heisenberg model with isotropic and finite-range interactions on a $d$-dimensional hypercubic lattice (c.f. Scholarpedia).

Broadly I am interested in whether it is possible to obtain results of this type (i.e. relating the existence of a phase transition to the dimension of the system) for models defined on more general graphs than regular lattices. For a graph $G$, one notion of dimension is the graph dimension, which roughly corresponds to the minimum dimension $d$ for which there is an embedding of the graph in $d$-dimensional Euclidean space such that all edges are of unit length.

To make this question more answerable, here is a specific question:

Is there a version of the Mermin-Wagner theorem for the quantum spin-$S$ Heisenberg model defined on a general graph $G$ (say with nearest-neighbour interactions), relating the existence of a phase transition to the graph dimension of $G$?

I would also be interested in answers which provide counterexamples to this idea.

Answer 1

This has essentially been answered here. Roughly speaking, the Mermin-Wagner theorem holds as long as the random walk on the graph is recurrent (see the answer I linked for a more precise setup; note that the paper by Merkl and Wagner I cite there also applies in the quantum case; see also this paper for a more recent closely related result).

In the particular case you are interested in, the answer to your question thus boils down to characterizing recurrence of the simple random walk on a graph in terms of other properties of the graph (spectrum, isoperimetry, various notions of dimension, etc.). There is a huge literature on this subject, with which I am only partially, and superficially, familiar. At this stage, the best would be to ask this new version of the question (that is, conditions on an infinite graph ensuring recurrence of the simple random walk) on MathOverflow, where you'll likely find experts.