Consider a nearest-neighbor percolation model on $\Bbb{Z}^d$, where each bond is occupied with probability $p$ independent of each other. Let $\Bbb{P}_p$ denote the corresponding law on the configuration space and $p_c$ the critical probability.

It is well-known that (under suitable conditions which I do not write here) the following limits give rise to the same probability measure, called the incipient infinite cluster (IIC) measure:[1]

  1. $\displaystyle \Bbb{P}_{\textsf{IIC}}(F) := \lim_{|x|\to\infty} \Bbb{P}_{p_c}(F \mid 0 \leftrightarrow x) $ for all cylindrical events $F$.

  2. $\displaystyle \Bbb{Q}_{\textsf{IIC}}(F) := \lim_{p \uparrow p_c} \frac{\sum_{x\in\Bbb{Z}^d} \Bbb{P}_p (F \cap \{ 0 \leftrightarrow x \}) }{\sum_{x\in\Bbb{Z}^d} \Bbb{P}_p (0 \leftrightarrow x) } $ for all cylindrical events $F$.

  3. $\displaystyle \Bbb{R}_{\textsf{IIC}} (F) := \lim_{r \to \infty} \Bbb{P}_{p_c} (F \mid 0 \leftrightarrow x \text{ for some } x \text{ with } |x| > r )$ for all cylindrical events $F$.


For me it seems reasonable to believe that IIC infinite clusters can be regarded as the infinite clusters which appear when $p$ is increased slightly from the critical phase $p_c$, or precisely,

$$ \Bbb{Q}_{\mathsf{IIC}}(F) = \lim_{p \downarrow p_c} \Bbb{P}_{p_c}(F \mid 0 \leftrightarrow \infty). $$

Indeed this was one of the original approaches when Kesten first constructed IIC measure on $\Bbb{Z}^2$. However, I was unable to find any reference for high-dimensional case so far. Is there any known progress related to this?


[1] M. Heydenreich, R. v. d. Hofstad, and T. Hulshof. High-dimensional incipient infinite clusters revisited. arXiv:1108.4325v2, (2012).


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