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Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with $n$ faces and boundary component lengths of order $\sqrt n$ or of lower order. Endow this quadrangulation with the usual graph metric renormalized by $n^{-1/4}$, mark it on each boundary component, and endow it with the counting measure on its vertex set renormalized by $n^{-1}$, as well as the counting measure on each boundary component renormalized by $n^{-1/2}$. We show that, as $n\to\infty$, this random marked measured metric space converges in distribution for the Gromov--Hausdorff--Prokhorov topology, toward a random limiting marked measured metric space called a Brownian surface. This extends known convergence results of uniform random planar quadrangulations with at most one boundary component toward the Brownian sphere and toward the Brownian disk, by considering the case of quadrangulations on general compact orientable surfaces. Our approach consists in cutting a Brownian surface into elementary pieces that are naturally related to the Brownian sphere and the Brownian disk and their noncompact analogs, the Brownian plane and the Brownian half-plane, and to prove convergence results for these elementary pieces, which are of independent interest.

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit., 76 pages, 29 figures, final version

We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map (CRUM) of a given genus that start with a suitably tilted Brownian continuum random tree and make `horizontal' point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth-first construction of a finite connected graph. In particular, this yields a breadth-first construction of the scaling limit of the critical Erd\H{o}s-R\'enyi random graph which answers a question posed in [2]. As a consequence of this breadth-first construction we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions., Comment: 39 pages, 7 figures; to appear in Random Structures & Algorithms

We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects., Comment: 91 pages, 12 figures

The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure. Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure. We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints. Finally, we classify the types of geodesic networks which are dense. For each $k\in\{1,2,3,4,6,9\}$, there is a dense set of pairs of points which are joined by networks of exactly $k$ geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense., 29 pages, 7 figures, final version

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the finite-range and the long-range setting. In the finite-range setting, this scaling limit is Brownian motion, in the long-range setting, it is a stable motion. The proof relies on a novel lace expansion that keeps track of the number of pivotal bonds., This is a thorough revision of a previous submission, arXiv:1301.3486v2, which was retracted because of an error in the proof. The results presented here are the same as those claimed in arXiv:1301.3486v2 (modulo a few details), but large parts of the proof are different. 86 pages, 46 figures

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_{c}$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p and polynomially if $p\geqslant p_{c}$.The critical percolation value is $p_{c}=1/2$ for site percolation, and $p_{c}=(2\sqrt{3}-1)/11$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_{c}$, the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

International audience; We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of • A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, • A property of invariance under re-rooting, • The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erd\"os-R\'enyi random graph. We exploit the explicit description of the scaling limit of the Erd\"os-R\'enyi random graph in the so-called critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST., Comment: 60 pages, 4 figures

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien (2015) by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense., 29 pages, 13 figures. Added reference and figure

We introduce and study the uniform infinite planar quadrangulation UIPQ with a boundary via an extension of the construction of Curien et al. Curien et al., Lat Am J Probab Math Stat 49 2013 340-373. We then relate this object to its simple boundary analog using a pruning procedure. This enables us to study the aperture of these maps, that is, the maximal graph distance between two points on the boundary, which in turn sheds new light on the geometry of the UIPQ. In particular we prove that the self-avoiding walk on the UIPQ is at most diffusive. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 30-58, 2015

We consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved., Comment: 18 pages, 2 figures

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the long- as well as in the finite-range setting. In the finite-range setting, this scaling limit is Brownian motion, in the long-range setting, it is a stable motion. The proof relies on a novel lace expansion for percolation that resembles the original expansion for self-avoiding walks by Brydges and Spencer in 1985. This expansion is interesting in its own right., Withdrawn because certain correction terms that arise in the Lace expansion of Section 3 were not identified and taken into account in the subsequent derivation. A new version with these correction terms included is in preparation

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., 76 pages, 7 figures, improved version

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambj��rn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

Consider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton-Watson tree with finite variance and conditioned to have size n, converges as $n\to\infty$ to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov-Prokhorov topology. This yields a multi-dimensional extension of a limit theorem due to Janson [Random Structures Algorithms 29 (2006) 139-179] for the number of random cuts needed to isolate the root in Galton-Watson trees conditioned by their sizes, and also generalizes a recent result [Ann. Inst. Henri Poincar\'{e} Probab. Stat. (2012) 48 909-921] obtained in the special case of Cayley trees., Comment: Published in at http://dx.doi.org/10.1214/12-AAP877 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\mathbb{R}$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.

These are the notes for a series of lectures given at the Clay Mathematical Institute Summer School in Buzios, July 11 - August 7, 2010. We review some of the recent aspects of scaling limits of random trees and planar maps, in particular via their relations with bijective enumeration and Gromov-Hausdorff convergence.

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 $\alpha\in(1,2)$. 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 $\alpha$. In particular, the profile of distances in the map, rescaled by the factor $n^{-1/2\alpha}$, 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 $n\to\infty$, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to $2\alpha$., Comment: Published in at http://dx.doi.org/10.1214/10-AOP549 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n\rightarrow \infty$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in $��$-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in our paper, Scaling limits of Markov branching trees, with applications to Galton--Watson and random unordered trees (2010)., Published in at http://dx.doi.org/10.3150/10-BEJ312 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

International audience; We prove that a uniform, rooted unordered binary tree with $n$ vertices has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees.

We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees., 25 pages, 2 figures

We give a new proof of a theorem by Le Gall & Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces., 11pp, 1 figure

We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic., 58pp, 6 figures. One figure added, minor corrections

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous's beta-splitting models and Ford's alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf., Comment: Published in at http://dx.doi.org/10.1214/07-AOP377 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some suitable moment assumptions., Comment: 44 pages

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., 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)

The basic object we consider is a certain model of continuum random tree, called the stable tree. We construct a fragmentation process $(F^-(t), t>=0)$ out of this tree by removing the vertices located under height $t$. Thanks to a self-similarity property of the stable tree, we show that the fragmentation process is also self-similar. The semigroup and other features of the fragmentation are given explicitly. Asymptotic results are given, as well as a couple of related results on continuous-state branching processes., 30 pages

We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2003) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of ``attracting points'' to emerge., 16 pages

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Levy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin., Comment: 32 pages

We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal H\"older exponents of the height functions.

We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the structure of an ICRT, and also of its width process, determining the size of layers in order of height. These processes turn out to be transformations of bridges with exchangeable increments, which have already appeared in other ICRT related topics such as stochastic additive coalescence. The results rely on two different constructions of birthday trees from processes with exchangeable increments, on weak convergence arguments, and on general theory on continuum random trees., Comment: 38 pages, 7 figures