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 $ϕ_{β_c}$ on $\mathbb Z^d$ with $d\geq 3$. We construct the measure $ϕ^\infty:=\lim_{|x|\rightarrow \infty}ϕ_{β_c}[\:\cdot\: |\: 0\leftrightarrow x]$ and prove it satisfies $ϕ^\infty[0\leftrightarrow \infty]=1$. 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 $χ(β)$ 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 $β<β_c$, \begin{equation*} χ(β)= \frac{A}{1-β/β_c}(1+o(1)), \end{equation*} where $o(1)$ tends to $0$ as $β$ tends to $β_c$. Additionally, we relate the constant $A$ to the incipient infinite cluster of the double random current.