On Loops in critical high-dimensional percolation

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).