J Bouttier and E Guitter 2009 J. Phys. A: Math. Theor. 42 465208 doi:10.1088/1751-8113/42/46/465208
J Bouttier and E Guitter
Show affiliationsWe consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulae for the bulk–boundary and boundary–boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian continuum random tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk–loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.
60J65 Brownian motion (See also 58J65)
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
Issue 46 (20 November 2009)
Received 29 June 2009, in final form 21 September 2009
Published 26 October 2009
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