Upcoming seminar:
Polyakov predicted in 1975 that the spin \(O(3)\) model (or classical Heisenberg model) should have exponential decay of correlations at all positive temperatures on \(\mathbb{Z}^2\). I will discuss the following result (j.w. with Christophe Garban). If we replace the Euclidean scalar product \(\langle s_i, s_j \rangle = s_i^x s_j^x + s_i^y s_j^y + s_i^z s_j^z\) by the \(\varepsilon\)-modified scalar product \(\langle s_i, s_j \rangle_\varepsilon := s_i^x s_j^x + s_i^y s_j^y + (1-\varepsilon) s_i^z s_j^z\), then for all \(1 \geq \varepsilon > 0\), the \(S^2\)-valued spin system with Gibbs measure proportional to \(\exp(\beta \sum_{i \sim j} \langle s_i, s_j \rangle_\varepsilon)\) has polynomial decay of correlations if \(\beta\) is chosen large enough.