What’s new in percolation theory?

Switching identities for percolation of Brownian loops on cable graphs and consequences

Speakers: Wendelin Werner

Date: Thursday, September 25th / 16:00h CEST

Abstract: It is known (since the work of Lupu) that if one samples a Poissonian cloud of Brownian loops on a cable graph according to some (natural) well-chosen intensity, the obtained clusters [the connected components of the union of the loops] have the very same law than the excursions away from 0 by a Gaussian Free Field on this cable graph. This can be viewed as one reason for which more detailed results end up being accessible for this model than for ordinary Bernoulli percolation.

We will discuss new results about this model, starting with a “switching identity” that allows to describe in simple terms the law of the clusters conditionally on two given points being connected. This has consequences in various directions, such as the “doubling property in high dimensions” (asymptotically, each cluster containing a very large cycle will asymptotically have a probability ½ of containing a very large Brownian loop) or a simple construction and description of the infinite incipient cluster for this model (in any dimension). We will also briefly mention some related results for ordinary percolation in high dimensions.

This talk is mainly based on https://arxiv.org/abs/2502.06754. We will also mention ongoing work with A. Carpenter and with T. Lupu.