What is known about some massive Gaussian models on a lattice?

Question

Recently I started to play with some massive Gaussian models on a lattice. Motivation being that I work on massless models and want to understand the massive case because it seems easier to handle (e.g. cluster expansion makes sense thanks to contours carrying mass). Consider a Hamiltonian

$$ H(x_{\Lambda}) = \sum_{<i,j> \in E(\Lambda)} (x_i - x_j)^2 +\sum_{i \in \Lambda} U(x_i) $$

with $\Lambda$ a lattice (think mainly of ${\mathbb Z}^d$ with $d = 2$), $E(\Lambda)$ the set of its edges, $x_{\Lambda} \in (\Lambda \to {\mathbb R})$ and $U(y)$ a potential.

I would like to know what is known about this class of models in general. E.g. for a related class of models without mass but with arbitrary two-point function Funaki and Spohn showed that there is no phase transition if that function is convex.

1. I wonder whether similar result is known for convex (replace with any other reasonable condition) $U$.

Similarly,

2. Is there a necessary condition on $U$ for there to be a phase transition? Try to give some examples.

E.g. it seems natural that there will be a transition for a symmetric double-well model (with spontaneous symmetry breaking at low temperatures) and one could also investigate the case with one of the wells being more favorable. I think I should be able to prove these things with some work but I suppose it's already been done by someone.

3. Could you point me to a reference about some double-well models?

Other kind of model I've been thinking about is (setting $\beta = 1$ and letting $p$ play the role of temperature)

$$\exp(-U(y)) = p \exp (-a y^2) + (1-p) \exp (-b y^2)$$ with $a$ suitably small and $b$ suitably large. This is a big well with a smaller well inside. Intuitively the system should sit in the small well at low temperatures ($p = 0$) and jump out and behave freely (with small mass $a$) at higher temperatures ($p = 1$), so this is a toy model of melting. Trouble is, I have no idea whether this really works and I can't decide whether there will or won't be a phase transition.

4. Any ideas about this model? Pointing me to a reference would be wonderful but I am not sure this has been studied before.

Answer 1

Unfortunately, I missed this question when it was asked... I don't know whether you're still looking for an answer, but just in case...

1. It depends what you mean by "there is no phase transitions". If you mean localization of the field in dimensions 1 and 2 (in these dimensions, the massless field having diverging variance), then this is (i) easy to do when U grows to infinity, and (ii) can even be done with an arbitrarily small negative bump at zero (this even yields exponential decay of correlations), see this paper for example. Of course, in all these case if you allow crazy boundary conditions, nasty things can happen.

2. Phase transition in such double-well potentials have been studied in several papers. Using relection positivity, you can look at this review paper. This has also been done using cluster expansions. I don't have references in mind, but there is a paper by Milos Zahradnik (ok, it seems to be this one)... You can also have a look at this paper and this one, and so many others...

3. This has been done: see this paper and its sequels (by Biskup and coauthors) and also the above mentioned review.
4. As you see, many things are known about that type of models. Let me also recommend this review of mine that covers this type of (and other) issues.

Answer 2

The reason you aren't getting answers here is not because this model is obscure. This model is the most extensively studied statistical model in history--- it is the lattice version of self-interacting scalar field theory. I suppose the reason you are having a hard time finding references for this particular thing is because much more is known by physicists than mathematical techniques are up to proving, but the standard literature on statistical fields are almost always talking about this model.

In addition to Kenneth Wilson's 1974 Reviews of Modern Physics article on Renormalization Group, I also like Parisi's little book "Statistical field theory", but there is also an encyclopedic treatment by Zinn-Justin, and there are textbooks at all levels. There are too many references to list.

The model has a renormalization group fixed point near 4 dimensions which is entirely described by the quadratic, cubic and quartic terms in the long-distance effective potential V. There are no other surviving terms--- any potential renormalizes to a quartic. This is also surprisingly true in 3 dimensions, where you might expect $\phi^6$ to contribute. It doesn't because the canonical dimension of the field is altered and makes this term irrelevant (this is discussed well in Parisi).

This is also the Ising model, because if you take the quartic term to be large, and the mass term to be large negative, you reproduce a two-state system at every site. The Ising model renormalizes to reproduce the general situation with a scalar field, so there is really no need to consider the full continuous field.

I could say more, but the field is enormous, and any summary is not going to be comprehensive or accurate. I would suggest to start with the Wikipedia page for Ising Model, to read Wilson, and Parisi. Also there is the classic papers by Symanzik on polymer models. You can ignore any rigorous work on this, both on the physics and mathematics end. None of it is useful because the proper rigorous framework is not yet available.