  The uniform infinite planar quadrangulation is an infinite random graph
embedded in the plane, which is the local limit of uniformly distributed finite
quadrangulations with a fixed number of faces. We study asymptotic properties
of this random graph. In particular, we investigate scaling limits of the
profile of distances from the distinguished point called the root, and we get
asymptotics for the volume of large balls. As a key technical tool, we first
describe the scaling limit of the contour functions of the uniform infinite
well-labeled tree, in terms of a pair of eternal conditioned Brownian snakes.
Scaling limits for the uniform infinite quadrangulation can then be derived
thanks to an extended version of Schaeffer's bijection between well-labeled
trees and rooted quadrangulations.
