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The two-dimensional Gaussian free field (GFF) is a random function defined on domains of the complex plane (more generally, Riemann surfaces). Just like Brownian motion is the basis for a theory of random paths, the GFF is the basis for a theory of random surfaces. However, contrary to Brownian motion, the GFF is almost surely not a continuous function; in fact, it is not even defined almost everywhere and only makes sense as a random distribution (in the sense of Schwartz).

The GFF lies at the crossroad between two different areas of probability theory. First, it is an instance of a logarithmically correlated Gaussian field; such fields arise commonly in random matrix theory, branching processes, stochastic models for the Riemann zeta-function and mathematical finance. Second, its law is conformally invariant: this implies deep connections with the representation theory of the Virasoro algebra, meaning that the GFF is amenable to algebraic tools.

The first part of the course will be devoted to the definition and main property of the GFF. In the second part, we will construct and solve Liouville conformal field theory, following (part of) the works of David, Guillarmou, Kupiainen, Rhodes and Vargas.

Prerequisites: The course will be essentially self-contained, with many reminders on Gaussian processes, stochastic and Malliavin calculus, distribution theory and conformal mapping. No prior knowledge of Lie or representation theory will be assumed.

Target audience: Students in probability theory with an interest in mathematical physics/algebra/integrability. Students with the converse profile are welcome but might find the probabilistic material challenging.

References

  • D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer.
  • D. Nualart. Analysis on the Wiener space. Springer Berlin Heidelberg (2006)
  • V. G. Kac, A. K. Raina. Bombay lectures on highest-weight representations of infinite dimensional Lie algebras. World Scientific
  • N. Berestycki and E. Powell. Gaussian free field, Liouville quantum gravity and Gaussian multiplicative chaos. https://homepage.univie.ac.at/nathanael.berestycki/Articles/master.pdf
  • R. Rhodes and Vincent Vargas. Gaussian multiplicative chaos and applications: a review. https://arxiv.org/abs/1305.6221
  • F. David, A. Kupiainen, R. Rhodes and V. Vargas. Liouville quantum gravity on the Riemann sphere. https://arxiv.org/abs/1410.7318
  • C. Guillarmou, A. Kupiainen, R. Rhodes and V. Vargas. Conformal bootstrap in Liouville theory. https://arxiv.org/abs/2005.11530
Semester: SuTerm 2022