Liouville quantum gravity spheres as matings of finite-diameter trees

Understand

We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways.

• The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder.
• The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path.
• In the special case that $\gamma=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable L\'evy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$.