On Loops in critical high-dimensional percolation

Abstract

We show the following results about critical Bernoulli percolation in high dimensions: In a box of side-length N, there exist self-avoiding open loops of diameter comparable to N, and the collection of these self-avoiding loops has a non-trivial scaling limit (if viewed in the Hausdorff topology) as N tends to infinity. This feature contrasts with the proliferation of "typical" percolation clusters pointed out by Michael Aizenman almost three decades ago. In other words, we show that among the many large clusters in a large box, only a handful will contain a self-avoiding loop of diameter greater than a fixed fraction of the side-length of the box.

38 pages, very much expanded version including full proofs and new results