Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1, . . . , n}, then with probability tending to 1 as n → ∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components

Convergence of Random Graphs

To trace back to the origin of the study of planar maps we have to go back to the ’60’s, when efforts to solve the 4-colour problem led to an increased interest in embedded planar graphs. Recently, being natural models for a random “discretised” surface, random pla- nar maps become relevant to theoretical physics, and in particular theories of quantum gravity. The question is how planar maps may in fact be interpreted as approximations of a “continuous” random surface. The aim of this work is to study the convergence of random graphs in local topology and scale limits, with a special look to planar maps. The major result we present, from papers of Le Galle and Miermont, is that scale limit of certain classes of planar maps is the Brownian Map. The work is divided in three chapters. In the first chapter we introduce the dis- crete objects and graph properties. We then introduce a metric in the space of pointed graph called the local topology and a notion of uniformly pointed graph through the mass transport principle. We show some examples of local limits in the topic of uniformly pointed graph with a particular attention to the local limit of Galton Watson trees. Then we briefly conclude with some connection to ergodic theory and percolation problems. In the second chapter we introduce the fundamental tools for the main result of the work. We start proving the Cori-Vanquelin-Shaeffer bijection between well labelled trees with n vertices and rooted quadrangulations with n faces. The proof of the CV S bijection follows the work of Shaeffer in his PhD thesis. We spend some efforts to study the metric properties preserved by the CV S bijection. After that we discuss the bijection with a well labelled embedded tree and his contour process. Inspired by the convergence of scaled random walks to the Brownian Motion (Donsker Theorem), we briefly introduce the theory of Brownian excursions and remarking the contour process method we connect large random plane trees and Brownian excursions. In the third chapter, we construct the Brownian Continuum Random Tree (CRT) and we see that with the help of contour process, CRT is the limit 1 of random planar tree. So, starting from the CRT we construct the Brownian Map. We conclude proving the result of Le Gall-Paulin, stating that the Brow- nian Map is homeomorphic to the 2-sphere and the result of Le Gall-Miermont, the Brownian Map is the limit of class of quadrangulations, in the Gromov- Hausdorff distance

Spanning forests and the vector bundle Laplacian

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.Comment: Published in at http://dx.doi.org/10.1214/10-AOP596 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The bead model and limit behaviors of dimer models

In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire; there must be exactly one bead on each neighboring wire. We construct a one-parameter family of Gibbs measures on the bead configurations that are uniform in a certain sense. When endowed with one of these measures, this model is shown to be a determinantal point process, whose marginal on each wire is the sine process (given by eigenvalues of large hermitian random matrices). We prove then that this process appears as a limit of any dimer model on a planar bipartite graph when some weights degenerate.Comment: Published in at http://dx.doi.org/10.1214/08-AOP398 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson-Voronoi tessellation in the $d$-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition $Z$ of these two processes, two points of $Z$ are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0-1 law, a subcritical phase as well as a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.Comment: 30 pages, 4 figures. Accepted for publication in Advances in Applied Probabilit