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

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 fo...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2009-11, Vol.42 (46), p.465208-465208 (44)
Main Authors: Bouttier, J, Guitter, E
Format: Article
Language:English
Subjects:
Combinatorics
Exact sciences and technology
Mathematical Physics
Mathematics
Physics
Probability
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Summary: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 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.
ISSN:1751-8121
1751-8113
1751-8121
DOI:10.1088/1751-8113/42/46/465208