Scaling limit of the cluster size distribution for the random current measure on the complete graph

TL;DR

This paper analyzes the scaling limit of cluster size distributions in the random current representation of the near-critical Ising model on the complete graph, connecting it to the switching lemma and tangling probabilities.

Contribution

It computes the scaling limit of cluster size distributions for arbitrary sources and relates tangling probabilities to the switching lemma in the Gaussian and Ising limits.

Findings

Derived the scaling limit of cluster size distribution.

Connected tangling probabilities with the switching lemma.

Revealed the Gaussian and Ising limits correspondence.

Abstract

We study the percolation configuration arising from the random current representation of the near-critical Ising model on the complete graph. We compute the scaling limit of the cluster size distribution for an arbitrary set of sources in the single and the double current measures. As a byproduct, we compute the tangling probabilities recently introduced by Gunaratnam, Panagiotis, Panis, and Severo in [GPPS22]. This provides a new perspective on the switching lemma for the $\varphi^4$ model introduced in the same paper: in the Gaussian limit we recover Wick's law, while in the Ising limit we recover the corresponding tool for the Ising model.