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

We consider the critical FK-Ising measure ${\phi }_{{\beta }_c}$ on ${\mathbb{Z}}^d$ with $d\ge 3$. We construct the measure ${\phi }^{\mathrm{\infty }}:={\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}{\phi }_{{\beta }_c}[\cdot \mathrm{\mid }0\leftrightarrow x]$ and prove it satisfies ${\phi }^{\mathrm{\infty }}[0\leftrightarrow \mathrm{\infty }]=1$. 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 $\chi (\beta )$ 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{equation*} χ(β)= \frac{A}{1-β/β_c}(1+o(1)), \end{equation*} 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.