Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

Details Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

2009-06-26

arxiv.org/abs/0906.4892v2

J. Phys. A: Math. Theor. 42 (2009) 465208

Physics

Mathematical Physics

55 pages, 14 figures, final version with minor corrections

Scientific paper

10.1088/1751-8113/42/46/465208

We 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 formulas 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.

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