Assign i.i.d. standard exponential edge weights to the edges of the complete graph Kn, and let Mn be the resulting minimum spanning tree. We show that Mn converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M . The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature [18], and contrast with the behaviour of invasion percolation on the PWIT [2] and on regular trees [6], which exhibit quadratic volume growth. 10 4. Properties of the forward maximal process ((X n , Z n ), n ≥ 1) 13 4.

76 pages, 7 figures, improved versionWe 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 → ∞ 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 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 geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α ∈ (1, 2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index α. In particular, the profile of distances in the map, rescaled by the factor n −1/2α , converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as n → ∞, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to 2α.

Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by $n^{1/4}$ to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight $q_k$ on faces of degree $2k$: the radius of such maps, conditioned to have $n$ faces (or $n$ vertices) and under a criticality assumption, converges in distribution once rescaled by $n^{1/4}$ to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton--Watson trees.Comment: Published in at http://dx.doi.org/10.1214/009117906000000908 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org