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Liouville quantum gravity spheres as matings of finite-diameter trees. (English. French summary) Zbl 1448.60168

Summary: 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 B. Duplantier in [“Liouville quantum gravity as a mating of trees”, Preprint, arXiv:1409.7055], 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 Lévy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by \(\mathrm{SLE}_6 \). This construction is relevant to a program for showing that the \(\gamma=\sqrt{8/3}\) Liouville quantum gravity sphere is equivalent to the Brownian map.

MSC:

60J67 Stochastic (Schramm-)Loewner evolution (SLE)
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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