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 work proves the existence of an incipient infinite cluster (IIC) measure for the FK-Ising model in dimensions d≥3 by connecting FK-Ising to the random current representation and employing a quantitative mixing property for currents. It provides two equivalent IIC constructions (via infinite-volume conditioning and a subcritical limit) and relates the IIC of the double random current to the susceptibility constant A(d) in high dimensions. For d>4, the authors establish the exact near-critical asymptotics χ(β) = A(d)/(1-β/β_c) (1+o(1)) as β↑β_c, and express A(d) in terms of IIC-avoidance events, with lim_{d→∞} A(d) = 1. The results offer a robust, lace-expansion-free route to IIC and near-critical behavior, highlighting deep connections between percolation-like IIC structures and the double random-current representation.

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.

Figures (4)

• Figure 1: A configuration $\mathbf{n}$ which satisfies the event appearing in \ref{['eq: proof thm 1 (3)']}. The dotted line highlights the fact that $0$ and $\mathbf{e}_1$ are not connected in the percolation configuration induced by $\mathbf{n}$.
• Figure 2: On the left, an illustration of a configuration realizing $\mathcal{B}_{u,v}$ for $u\notin \Lambda_m$. The backbone of $\mathbf{n}_1$ (resp. $\mathbf{n}_2$) is the black (resp. red) bold line. A string of loops connects $\Gamma(\mathbf{n}_2)$ to $\Gamma(\mathbf{n}_1)$ in $\mathbf{n}_1+\mathbf{n}_2$ without using any edges of $\overline \Gamma(\mathbf{n}_1)$. On the right, a diagrammatic representation of the bound obtained in \ref{['eq: bound u not in lambda m']}.
• Figure 3: An illustration of the events $\mathcal{G}_1$ and $\mathcal{G}_2$. The backbone of $\mathbf{n}_1$ (resp. $\mathbf{n}_2$) is the black (resp. red) bold line. On the left, $\Gamma(\mathbf{n}_2)$ crosses $\text{Ann}(M,k)$ twice and a string of loops creates a connection between the clusters of $0$ and $\mathbf{e}_1$. On the right, $\Gamma(\mathbf{n}_1)$ and $\Gamma(\mathbf{n}_2)$ only cross $\text{Ann}(M,k)$ once. This forces the existence of a crossing of $\textup{Ann}(m,M)$ in $\mathbf{n}_1+\mathbf{n}_2\setminus \overline\Gamma(\mathbf{n}_1)\cup\overline\Gamma(\mathbf{n}_2)$.
• Figure 4: An illustration of a configuration $\mathbf{n}_i'$ contributing to the event $F_i$ introduced in \ref{['eq: def F_i']}. The backbone of $\mathbf{n}_i'$ is the bold black line. Here, $\Gamma(\mathbf{n}_i')$ does not cross $\textup{Ann}(r,m)$ but $\mathbf{n}_i'$ crosses $\textup{Ann}(n,m)$ thanks to a string of loops crossing $\textup{Ann}(n,r)$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture
• Definition 2.1
• Definition 2.2
• Lemma 2.3: Switching Lemma
• Remark 2.4