  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. We give exact formulae based on the DOZZ formula in
the Liouville case and asymptotic behaviours in the other cases, with a
probabilistic representation of the limit.
  Finally, we show how to extend our results to boundary LCFT, treating the
cases of the fusion of two boundary or bulk insertions as well as the
absorption of a bulk insertion on the boundary.
