in Lectures on the Spin and Loop *O*(*n*) Models , Ron Peled and Yinon Spinka say :

Polyakov predicted in 1975 that the spin O(n) model with n>=3 should exhibit exponential decay of correlations in two dimensions at any positive temperature. ... ... **Giving a mathematical proof of this statement (or its analog in infinite volume) remains one of the major challenges of the subject.** The best known results in this direction are by Kupiainen who performed a 1=n-expansion as n tends to infinity.

Antti J. Kupiainen Comm. Math. Phys. 73 (1980), no. 3, 273-294

Abstract. The ί/n expansion is considered for the π-component non-linear σ-model (classical Heisenberg model) on a lattice of arbitrary dimensions. We show that the expansion for correlation functions and free energy is asymptotic, for all temperatures above the spherical model critical temperature. Furthermore, the existence of a mass gap is established for these temperatures and n sufficiently large.

The § *2.2 Main results and conjectures* in the first link is a good review and might complete the wiki page.