The CRT is the scaling limit of random dissections. Issue 2 (11th June 2014)

• Record Type: Journal Article
• Authors: Curien, Nicolas; Haas, Bénédicte; Kortchemski, Igor
• Is Part Of: Random structures & algorithms. Volume 47:Issue 2(2015)
• Page Start: 304
• Page End: 327
• Publication Date: 2014-06-11
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20554
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>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 <italic>p</italic>‐angulations. As their number of vertices <italic>n</italic> goes to infinity, we show that these random graphs, rescaled by <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twp0n5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20554:rsa20554-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>, 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</p> </abstract>