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

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.

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No one wants this bounty? –  Larian LeQuella Jan 10 '12 at 11:34
I'd guess that nobody with the expertise to answer this has seen the question. –  David Z Jan 11 '12 at 0:26

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.
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BTW, if you'd like more informations, or additional references, etc., you should rather contact me by email (the address is indicated on my homepage). –  Yvan Velenik Jun 1 '12 at 18:17
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).