Spins, loops and fields

Random height functions as models for random surfaces play an important role in statistical mechanics. As direct generalisations of random walks to “two-dimensional” time, they are also fundamental objects from the point of view of modern probability theory. In vast generality, their large-scale fluctuations are predicted to be described by the continuum field called the Gaussian free field (GFF). This universal random object is a model of a free bosonic field theory and is the centrepiece around which many available mathematical descriptions of conformal field theories are built. This prediction has been so far mathematically established only in a handful of cases exhibiting some forms of ex- act solvability. The current project aims at deepening our understanding of the geometric structures underlying many natural discrete models of height functions as well as the continuum GFF.

Recent fascinating developments show that the GFF, even though it is not a random surface defined pointwise, but rather a distribution acting on test functions, has a rich structure of built-in geometric objects that may be thought of as the level lines of the field. The proposed project brings out a number of new and precise conjectures predicting that certain discrete interfaces arising in a classical model of statistical mechanics, the Ashkin–Teller model, converge in the scaling limit to the level lines of the GFF. We aim at establishing these conjectures, and understanding in detail the parallels between the discrete and continuum geometric descriptions of height functions. An important part of this program is to give a probabilistic construction of the purported continuum Ashkin–Teller correlation functions as true expectations with respect to random fields naturally defined through the GFF’s level lines.

The project also proposes to study the classical XY -model. This spin model with abelian continuous symmetry famously undergoes the Berezinskii–Kosterlitz–Thouless (BKT) phase transition in two dimensions. For this discovery, Kosterlitz and Thouless were awarded the 2016 Nobel prize in physics. Recently a new geometric representation of the model was introduced in form of a random collection of loops. It was used, together with the analysis of the associated height function model, to derive a new and highly simplified proof of the BKT transition that opens new directions of research. Our aim is to understand better the two-way relationship between the localisation-delocalisation phase transition in the height function model, and the celebrated BKTs phase transition in the spin model (they should happen at the same temperature). Another suggested application is to use the loop representation and its switching lemmas to approach questions like continuity of phase transition in this model.

We will also investigate the fascinating phenomenon of bosonisation of fermions, and fermionisation of bosons in two-dimensional statistical mechanics. In parallel to the celebrated bosonisation identities of Dubédat for the Ising model, we propose to construct (approximations of) Kadanoff–Ceva fermions in the discrete Gaussian free field. The aim is to use these observables to analyze crossing probabilities of level lines of the continuum GFF. We also propose to study Green’s functions of a particular type of level lines – the multiple SLE4 curves – using the (still conjectural) relation with the double dimer model.