We consider the critical FK-Ising measure \(\phi _{\beta _c}\) on \(\mathbb Z^d\) with \(d\ge 3\) . We construct the measure and prove it satisfies . 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 \(\chi (\beta )\) of the nearest-neighbour Ising model on \(\mathbb Z^d\) . When \(d>4\) , we improve a previous result of Aizenman (Comm. Math. Phys., 1982) to obtain the existence of \(A>0\) such that, for \(\beta <\beta _c\) , \(\begin{aligned} \chi (\beta )= \frac{A}{1-\beta /\beta _c}(1+o(1)), \end{aligned}\) where o(1) tends to 0 as \(\beta \) tends to \(\beta _c\) . Additionally, we relate the constant A to the incipient infinite cluster of the double random current.
