Phase diagram of the loop O(n) model

Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models. The focus of this talk is on the loop O(n) model on the hexagonal lattice. Among its particular cases are percolation, Ising model, self-avoiding walk, dimers, integer-valued Lipschitz functions, proper 4-colourings and others. The loop O(n) model has attracted a lot of attention due to its rich phase diagram that includes a large region of parameters with an expected conformally invariant scaling limit. The model is difficult to study due to the lack of monotonicity with respect to parameters: each point of the phase diagram should be treated separately. In particular, existence of macroscopic loops has been established only recently in several sparse regions of parameters. I will present a unifying approach that applies to all n between 1 and 2. Main tools: novel graphical representation, Benjamini—Schramm limit, XOR argument (aka arXiv:2001.11977) and a new bound on a percolation threshold (due to Harel and Zelesko). This is joint work with Matan Harel (Northeastern).