#paperOfTheDay : "The Brownian loop soup" from 2003. This is a #mathematics paper about random walks in the plane, but there is a famous concrete example from #physics : the 2-dimensional Brownian motion. Generically, the trajectory of such a random walk in a plane can intersect itself. A special case are the non-intersecting random walks. They can be obtained by taking a self-intersecting one, and whenever it intersects itself, one discards the "loop" part (shown in blue in my drawing). What remains is a random walk without self intersection (shown in black).
The purpose of the present paper is to prove that this situation has an equivalent second interpretation: One can view it as a non-intersecting random walk that proceeds through a "Brownian loop soup", namely the collection of random self-intersecting detached loops. Whenever the walk intersects a loop for the first time (green dot), one glues in this loop, and thereby obtains a self-intersecting walk. The non-trivial proof is that this is not just "similar", but in fact mathematically equivalent: The properties of distributions are identical regardless if one starts from the loop soup, or from a self-intersecting walk. #dailyPaperChallenge https://arxiv.org/abs/math/0304419