On Loops in critical high-dimensional percolation

TL;DR

This paper studies the structure of large clusters in high-dimensional critical percolation, showing that clusters with large loops are rare and their scaling limits relate to Brownian loop-soups, contrasting with typical cluster behavior.

Contribution

It introduces a detailed analysis of large loop-containing clusters in high-dimensional critical percolation and connects their scaling limits to Brownian loop-soups, highlighting their rarity.

Findings

Clusters with large loops are tight in high dimensions.

Large loop-containing clusters have scaling limits related to Brownian loop-soups.

Such clusters are rare compared to typical percolation clusters.

Abstract

We discuss the following type of results about critical Bernoulli percolation in high dimensions: The collection of clusters that do contain large (self-avoiding) loops in a large box is tight. The collection of these large loops has scaling limits that one will be able to relate to Brownian loop-soups (each of these atypical loop-containing clusters does in some sense only contain one large self-avoiding loop in the sense that any two such loops will be very close). This feature contrasts with the known proliferation of "typical" percolation clusters (i.e., among the many large clusters in a given box, only a handful will contain a loop of size comparable to the box).