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

TL;DR

This paper constructs the incipient infinite cluster measure for the FK-Ising model in dimensions $d extgreater=3$ and analyzes the susceptibility of the high-dimensional Ising model, revealing new properties and precise asymptotics.

Contribution

It introduces a measure for the incipient infinite cluster in the FK-Ising model and refines the understanding of susceptibility in high-dimensional Ising models, connecting these concepts through advanced probabilistic techniques.

Findings

Construction of the incipient infinite cluster measure for FK-Ising in $d extgreater=3$

Proof that the measure satisfies $ ext{probability of infinite cluster}=1$

Asymptotic formula for susceptibility $oxed{ ext{for } d>4}$

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.