Editing Classical XY model (section)

===Three and higher dimensions===
Independently of the range of the interaction, at low enough temperature the magnetization is positive.

* At high temperature, the spontaneous magnetization vanishes: <math> M(\beta):=|\langle  \mathbf{s}_i\rangle|=0 </math>. Besides, [[cluster expansion]] shows that the spin correlations cluster exponentially fast: for instance <math> |\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle| \le C(\beta)e^{-c(\beta)|i-j|} </math>.
* At low temperature, [[infrared bound]] shows that the spontaneous magnetization is strictly positive: <math> M(\beta):=|\langle \mathbf{s}_i\rangle|>0</math>. Besides, there exists a 1-parameter family of extremal states, <math> \langle \; \cdot \; \rangle^\theta</math>, such that <math> \langle \mathbf{s}_i\rangle^\theta= M(\beta) (\cos \theta, \sin \theta) </math> but, conjecturally, in each of these extremal states the truncated correlations decay algebraically.