The CRT is the scaling limit of random dissections. Issue 2 (11th June 2014) - Curien, Nicolas; Haas, Bénédicte; Kortchemski, Igor
<x xml:space="preserve">Abstract</x> Abstract–We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform p‐angulations. As their number of vertices n goes to infinity, we show that these random graphs, rescaled by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twp0n5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />n−1/2</alternatives>, converge in the Gromov–Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 304–327, 2015