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

Abstract: We consider the critical FK-Ising measure $\phi_{\beta_c}$ on $\mathbb Z^d$ with $d\geq 3$. We construct the measure $\phi^\infty:=\lim_{|x|\rightarrow \infty}\phi_{\beta_c}[:\cdot: |: 0\leftrightarrow x]$ and prove it satisfies $\phi^\infty[0\leftrightarrow \infty]=1$. 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*} \chi(\beta)= \frac{A}{1-\beta/\beta_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.