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.