Related papers: The CRT is the scaling limit of random dissections

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…

We prove that Aldous' Brownian CRT is the scaling limit, with respect to the Gromov--Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected…

A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series-parallel maps…

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…

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees…

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by…

We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the…

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…

We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these…

In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the…

We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube…

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…

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As…

We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and…

We prove that the uniform unlabelled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This proves a conjecture by…

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the…

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit…

To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the $k$-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved…

We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree…