"Gaussian free field and Schramm--Loewner Evolution" (Part 2)

Eveliina Peltola
Aalto University
2nd affiliation: University of Bonn, Bonn, Germany

We introduce two canonical objects in random geometry: the Gaussian free field (GFF), aka the free boson, and the Schramm--Loewner Evolution (SLE).

The GFF is a rough version of a random harmonic generalized function. In two dimensions it (conjecturally) describes the scaling limit of the height function in many statistical physics models, such as the dimer and double dimer models, and the six vertex model. The GFF also describes the quantum field theory of a free boson and is a crucial ingredient in the rigorous construction of Liouville conformal field theory in two dimensions.

Schramm--Loewner Evolution (SLE) is a family of random curves characterised by conformal invariance and a domain Markov property, which makes them canonical candidates for interfaces in models of statistical mechanics at their critical point.

The tutorial will present a basic introduction to these two objects. We will also try to highlight some of the many intricate connections linking these two objects. For instance, the "zero-height" level lines of a GFF are SLE curves with parameter \kappa = 4.

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