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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.

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Note that the claim "a random quadrangulation converges in the Gromov-Hausdorff metric to a limiting metric space" is currently only known to be true along a suitable subsequence (though perhaps someone with more current knowledge will come by and correct me). This subtlety is what I was hinting at below. –  j.c. Nov 5 '10 at 11:14
    
@jc: Yes, you are right, I should have hedged my summary. Thanks. –  Joseph O'Rourke Nov 5 '10 at 11:59
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I'm not sure if this is what is being referred to, but there is a notion of random planar maps which are random triangulations of the sphere and the "scaling limit" of large triangulations is expected to yield a random metric space with the topology of $S^2$ (called the "Brownian Map") though this has not yet been completely proved. Various funny results are known though, such as that $S^2$ with this random metric has Hausdorff dimension 4.

The webpage of Le Gall contains lecture notes around these ideas. See particularly this recent introduction. Olivier Bernardi also has pretty slides for two lectures on the topic "Scaling Limit of Random Planar Maps".


Schaeffer

update of April 10, 2011: Grégory Miermont has posted a preprint claiming a proof of the result I was vaguely describing above for quadrangulations, rather than triangulations (see the final section of his preprint for thoughts on "universality"). According to the introduction, Jean-François Le Gall has also proven a similar result with a different method recently as well. The abstract:

We prove that uniform random quadrangulations of the sphere with $n$ faces, endowed with the usual graph distance and renormalized by $n^{-1/4}$, converge as $n\to\infty$ in distribution for the Gromov-Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called {\em Brownian map}, which was introduced by Marckert & Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of {\em geodesic stars} in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

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Even if this is not the source, LeGall's and Bernardi's work is fascinating! I especially enjoy Shaeffer's bijection between trees and quadrangulation maps. I took the liberty of adding an image from LeGall's lectures (due to Shaeffer) to your post. Thanks! –  Joseph O'Rourke Nov 4 '10 at 11:23
    
I should add that this is (conjecturally) related to a model of random geometry studied by physicists called Liouville Quantum Gravity which is then connected to the Gaussian Free Field (roughly speaking, the 2D version of Brownian motion). Check out the course material of Scott Sheffield on this page claymath.org/programs/summer_school/2010 and other papers by him and Bertrand Duplantier. –  j.c. Nov 4 '10 at 14:07
    
@jc: Thanks for pointing out these courses. I was ignorant of this line of work, but the GFF is very interesting and relevant to my concerns. Great! –  Joseph O'Rourke Nov 4 '10 at 20:58
    
The paper of Le Gall's is now out on the arXiv: arxiv.org/abs/1105.4842 and apparently covers the case of q-angulations for all even $q\geq4$ and q=3 –  j.c. May 25 '11 at 1:57
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