Liouville field theory is the model for two-dimensional quantum gravity. It was constructed rigorously using probabilistic methods by David-Kupiainen-Rhodes-Vargas in 2016. According to the conformal bootstrap conjecture n-point correlation functions can be expressed in terms of 3-point correlation functions and so-called conformal blocks.
We restrict ourselves to the case of one point correlation function of the Liouville field theory on the torus. We want to study conformal blocks. They are described using complicated asymptotic series. The probabilistic model for them was suggested by Ghosal-Remy-Sun-Sun in 2021. It allowed showing that the asymptotic series is actually converging in a small disc.
Liouville field theory has central charge c associated to it. Zamolodchikov in 1984 conjectured that conformal blocks have a limit as c goes to infinity. The limit was called classical conformal blocks. We use the probabilistic formula for conformal blocks to prove Zamolodchikov conjecture and show that the asymptotic series for them is converging in a small disc.
This is joint work with Harini Desiraju and Promit Ghosal.