Scaling Limits for Random Quadrangulations of Positive Genus

Details Scaling Limits for Random Quadrangulations of Positive Genus Scaling Limits for Random Quadrangulations of Positive Genus

2010-02-19

arxiv.org/abs/1002.3682v3

Electronic Journal of Probability 15, 52 (2010) 1594-1644

Mathematics

Probability

Scientific paper

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter.

Affiliated with

Also associated with

No associations

Rating

Scaling Limits for Random Quadrangulations of Positive Genus does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Scaling Limits for Random Quadrangulations of Positive Genus, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Scaling Limits for Random Quadrangulations of Positive Genus will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-363391