A note on the Ising model in high dimensions (Q1115043)

We consider the d-dimensional Ising model with a nearest neighbor ferromagnetic interaction \(J(d)=1/4d\). We show that as \(d\to \infty\) the \(+\) phase (and the - phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external field \(h\neq 0\).