Statistics of geodesics in large quadrangulations
I consider the statistical properties of geodesics, i.e. paths of
minimal length, in large random planar quadrangulations. Through an
extension of Schaeffer's well-labeled tree construction, I will obtain
expressions for the generating functions of planar quadrangulations with
a marked geodesic, as well as with a set of "confluent geodesics", i.e.
a collection of non-intersecting geodesics connecting two given points.
I will then explain how these results translate into the mean number of
geodesics in a large random quadrangulation, and other statistical
averages.