Dimensional Phase Transition in Random Walk Loop Soup

We consider a system of closed random walk trajectories interacting by mutual repulsion. This probabilistic model is motivated by its connections to statistical mechanics models, such as the Bose gas, the double dimer model, or the spin O(N) model. We prove the occurrence of macroscopic loops in dimension d>2 (joint with A. Quitmann, 2022) and the absence of macroscopic loops in dimension d=2 (joint with W. Wu, 2024). The talk will primarily focus on the two-dimensional case, which is analyzed using a novel complex spin representation and the derivation of a Mermin-Wagner theorem for complex measures.