The incipient infinite cluster of the FK-Ising model in dimensions d≥ 3 and the susceptibility of the high-dimensional Ising model

We consider the critical FK-Ising measure φβc on Zᵈ with d≥ 3. We construct the measure φ^∞:=lim|x|→ ∞φβc[\:·\: |\: 0↔ x] and prove it satisfies φ^∞[0↔ ∞]=1. This corresponds to the natural candidate for the incipient infinite cluster measure of the FK-Ising model. Our proof uses a result of Lupu and Werner (Electron. Commun. Probab., 2016) that relates the FK-Ising model to the random current representation of the Ising model, together with a mixing property of random currents recently established by Aizenman and Duminil-Copin (Ann. Math., 2021). We then study the susceptibility χ(β) of the nearest-neighbour Ising model on Zᵈ. When $d>4$, we improve a previous result of Aizenman (Comm. Math. Phys., 1982) to obtain the existence of $A>0$ such that, for β<βc, {equation*} χ(β)= A/1-β/β_c(1+o(1)), {equation*} where $o(1)$ tends to $0$ as β tends to βc. Additionally, we relate the constant A to the incipient infinite cluster of the double random current.