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 $d\geq 3$ 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 $d\geq 5$. 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 $d\geq 4$, 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 $\eta\leq 1/2$. As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when $d=3,4$.