MATH 506, The Gaussian Free Field


There are many connections between the mathematical theory of probability, statistical physics, and quantum field theory. One topic which has attracted a lot of recent attention is the Gaussian free field (GFF). GFF is a model for noninteracting particles, and interaction terms can be added later. GFF also gives a model for random surfaces. On the mathematical side, GFF appears as a limiting object for a wide variety of random processes. Making rigorous sense of interacting models presents some interesting challenges, since one obtains infinite terms which must be removed by renormalization. Such interacting models also arise as stationary distributions for stochastic PDE.

The mathematical basis of GFF rests on Gaussian processes, so we will start by briefly covering the relevant theory. The rest of the course will be based on various online lecture notes, as well as current literature.

The main prerequisite is some knowledge of measure-theoretic probability, such as Math 403.

Time and Place

  • Tuesdays and Thursdays 2:00 pm - 3:15 pm
  • LeChase 163


Each student will give a lecture later in the semester.



Introduction and Overview (First week or two)

  • GFF as a multidimensional generalization of Brownian motion.
  • Brief overview of the Euclidean approach to Feynman integrals and quantum field theory, the importance of moments.
  • Gaussian vectors, also called jointly Gaussian random variables.
  • Construction of the canonical Gaussian process over a Hilbert space with a countable basis.

Werner and Powell’s Notes

  • Properties of the discrete Gaussian Free Field.
  • Covariance given by Green’s function
  • Spatial Markov property
  • Determinant of the Laplacian
  • Uniform spanning trees
  • Loop soups