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Register a new userDistance statistics in quadrangulations with no multiple edges and the geometry of minbus
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Added by
J\\'er\\'emie Bouttier
Publication date
2010
fields
Physics
and research's language is
English
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We present a detailed calculation of the distance-dependent two-point function for quadrangulations with no multiple edges. Various discrete observables measuring this two-point function are computed and analyzed in the limit of large maps. For large distances and in the scaling regime, we recover the same universal scaling function as for general quadrangulations. We then explore the geometry of minimal neck baby universes (minbus), which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges, the mother universe. We give a number of distance-dependent characterizations of minbus, such as the two-point function inside a minbu or the law for the distance from a random point to the mother universe.
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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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