I heard it claimed that there is, in some sense, only
one *random metric* on $\mathbb{S}^2$.
I would appreciate any pointer to literature that explicates
this intriguing claim.
So far my own searches have not struck a suitable source.
I don't know enough about the topic to even define
what constitutes a random metric in the context of this claim,
so I cannot ask a sharper question.
I seek references to learn more. Thanks!

**Addendum**.
Reading the literature kindly suggested by jc, I believe that the source is the work
of
Jean-Francois Le Gall, and in particular, his paper
"The topological structure of scaling limits of large planar maps"
*Invent. Math.* **169** (2007), no. 3, 621--670
arXiv:math/0607567v2 math.PR.
He shows that a random quadrangulation converges in the Gromov-Hausdorff metric
to a limiting metric space.
Here is a quote from Le Gall's lectures at Clay Inst. on the topic:

This limiting random metric space, which is called the Brownian map, can be viewed as a "Brownian surface" in the same sense as Brownian motion is the limit of rescaled discrete paths. The Brownian map is almost surely homeomorphic to the two-dimensional sphere, although it has Hausdorff dimension 4.