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

Understand

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.