Abstract: 

A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle. The loop O(n) model on the hexagonal lattice is a random loop configuration, with the energy of of a loop configuration taken to be linear in the number of edges and the number of loops. I will discuss the resulting phase structure of the loop O(n) model, focusing on recent results about the non-existence of macroscopic loops for large n, and about the existence of macroscopic loops on a critical line when n is between 1 and 2. 

Talk based on joint works with Hugo Duminil-Copin, Alexander Glazman, Ron Peled and Wojciech Samotij.