What is this model? [closed]

Question

Consider the following model in classical statistical mechanics. Take a finite box $\Lambda\subseteq\mathbb{Z}^d$ and consider the field $\phi:\Lambda\to[-1,1]$ whose Gibbs measure is given by $$ \mathrm{d}\mathbb{P}(\phi):=\frac{\exp\left(\frac{1}{2}\beta\sum_{x\in\Lambda}\sum_{y\sim x}\phi_x\phi_y\right)}{\left(\prod_{x\in\Lambda}\int_{\phi_x\in[-1,1]}\mathrm{d}\phi_x\right)\exp\left(\frac{1}{2}\beta \sum_{x\in\Lambda}\sum_{y\sim x}\phi_x\phi_y\right)}\prod_{x\in\Lambda}\mathrm{d}\phi_x\,. $$

1. Does this model have a name? Has it been studied?
2. I am trying to ascertain if this model has a phase transition in various dimensions $d$, i.e., if there is some $\beta_c<\infty$ such that above it the two point function $$ \mathbb{E}\left[\phi_x\phi_y\right] $$ is polynomial decaying or constant and below it, it is exponentially decaying. Has this been studied? Is there a simple argument, e.g., for the non-existence of a phase transition in $d=2$?

Answer 1

This is the continuous spin Ising model. It has a phase transition at large $\beta$ in any dimension greater or equal to $2$; see this paper, for instance.

It is also known as the infinite spin Ising model. It was first introduced by Griffiths in this paper.

See also this paper for a more geometrical way of analyzing this model (through a generalization of the Fortuin-Kasteleyn random cluster representation).