Liouville quantum gravity spheres as matings of finite-diameter trees
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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
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Engineering and Physical Sciences Research Council (EP/L018896/1)