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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2009

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

Résumé

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.

Dates et versions

hal-00586654 , version 1 (18-04-2011)

Identifiants

Citer

Jérémie Bouttier, Emmanuel Guitter. Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. Journal of Physics A: Mathematical and Theoretical, 2009, 42 (46), pp.465208. ⟨10.1088/1751-8113/42/46/465208⟩. ⟨hal-00586654⟩
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