New lower bounds for the (near) critical Ising and φ⁴ models' two-point functions

We study the nearest-neighbour Ising and φ⁴ models on Zᵈ with d≥ 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≥ 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 η=0 and ν=1/2 when d≥ 4, where η is the critical exponent associated with the decay of the model's two-point function at criticality and ν is the critical exponent of the correlation length ξ(β). When $d=3$, we improve previous results and obtain that η≤ 1/2. As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when $d=3,4$.