1 research outputs found
The topological structure of scaling limits of large planar maps
We discuss scaling limits of large bipartite planar maps. If p is a fixed
integer strictly greater than 1, we consider a random planar map M(n) which is
uniformly distributed over the set of all 2p-angulations with n faces. Then, at
least along a suitable subsequence, the metric space M(n) equipped with the
graph distance rescaled by the factor n to the power -1/4 converges in
distribution as n tends to infinity towards a limiting random compact metric
space, in the sense of the Gromov-Hausdorff distance. We prove that the
topology of the limiting space is uniquely determined independently of p, and
that this space can be obtained as the quotient of the Continuum Random Tree
for an equivalence relation which is defined from Brownian labels attached to
the vertices. We also verify that the Hausdorff dimension of the limit is
almost surely equal to 4.Comment: 45 pages Second version with minor modification