Fusion asymptotics for Liouville correlation functions

Understand

David-Kupiainen-Rhodes-Vargas introduced a probabilistic framework based on the Gaussian Free Field and Gaussian Multiplicative Chaos in order to make sense rigorously of the path integral approach to Liouville Conformal Field Theory (LCFT).

• We use this setting to compute fusion estimates for the four-point correlation function on the Riemann sphere, and find that it is consistent with predictions from the framework of theoretical physics known as the conformal bootstrap.
• This result fits naturally into the famous KPZ conjecture which relates the four-point function to the expected density of points around the root of a large random planar map weighted by some statistical mechanics model.
• From a purely probabilistic point of view, we give non-trivial results on negative moments of GMC.