New Lower Bounds for the (Near) Critical Ising and \(\varphi ^4\) Models’ Two-Point Functions

We study the nearest-neighbour Ising and \(\varphi ^4\) models on \({\mathbb {Z}}^d\) with \(d\ge 3\) and obtain new lower bounds on their two-point functions at (and near) criticality. Together with the classical infrared bound, these bounds turn into up to constant estimates when \(d\ge 5\). When \(d=4\), we obtain an “almost” sharp lower bound corrected by a logarithmic factor. As a consequence of these results, we show that \(\eta =0\) and \(\nu =1/2\) when \(d\ge 4\), where \(\eta \) is the critical exponent associated with the decay of the model’s two-point function at criticality and \(\nu \) is the critical exponent of the correlation length \(\xi (\beta )\). When \(d=3\), we improve previous results and obtain that \(\eta \le 1/2\). As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when \(d=3,4\).