Random Quadrangulations Research Articles

The SLE loop via conformal welding of quantum disks

We prove that the SLEκ loop measure arises naturally from the conformal welding of two γ-Liouville quantum gravity (LQG) disks for γ2=κ∈(0,4). The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE8∕3 loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.

Planarity and non-separating cycles in uniform high genus quadrangulations

We study large uniform random bipartite quadrangulations whose genus grows linearly with the number of faces. Their local convergence was recently established by Budzinski and the author [9, 10]. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar with high probability up to logarithmic radius, and we prove that there exist non-contractible cycles of constant length with positive probability.

Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW. Our proof uses only the peeling procedure for random quadrangulations and some basic properties of the Brownian half-plane, so can be read without any knowledge of SLE or LQG.

The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model

We study the branching tree of the perimeters of the nested loops in the non-generic critical O(n) model on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution (x_i)_{i \geq 1} is related to the jumps of a spectrally positive \alpha -stable Lévy process with \alpha= \frac{3}{2} \pm \frac{1}{\pi} \mathrm {arccos}(n/2) and for which we have the surprisingly simple and explicit transform \mathbb{E}\Big[ \sum_{i \geq 1}(x_i)^\theta \Big] = \frac{\sin(\pi (2-\alpha))}{\sin (\pi (\theta - \alpha))}, \quad for \theta \in (\alpha, \alpha+1). An important ingredient in the proof is a new formula of independent interest on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time. We also identify the scaling limit of the volume of the critical O(n) -decorated quadrangulation using the Malthusian martingale associated to the continuous multiplicative cascade.

Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal $\mathrm{SLE}_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane $\mathrm{SLE}_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane $\mathrm{SLE}_{8/3}$. Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-LQG.

Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

Nous démontrons que la quadrangulation de Boltzmann libre avec un bord simple de périmètre fixé, munie de sa métrique de graphe, de sa mesure d’aire naturelle, et du chemin qui décrit sa frontière, converge dans la limite d’échelle vers le disque brownien libre de Boltzmann. La topologie de cette convergence est celle de Gromov–Hausdorff–Prokhorov-uniforme (GHPU), qui est l’analogue naturel de la topologie de Gromov-Hausdroff pour des espaces métriques mesurés décorés par une courbe. Nous déduisons de cela qu’une quadrangulation aléatoire de la sphère, décorée par une marche aléatoire auto-évitante de longueur $2l$, converge en loi pour la topologie GHPU vers l’espace métrique mesuré et décoré par une courbe que l’on obtient en recollant ensemble deux disques browniens indépendants le long de leurs bords.

Joint convergence of random quadrangulations and their cores

Nous montrons qu’une quadrangulation uniformément aléatoire, sa plus grande composante 2-connexe, et sa plus grande composante simple convergent conjointement en loi vers la même carte brownienne dans le sens Gromov–Hausdorff–Prokhorov. En premier, nous étendons l’analyse de (Random Structures Algorithms 19 (2001) 194–246) afin de démontrer un théorème limite local pour les tailles des plus grandes composantes. Les bornes sur les diamètres ainsi obtenues impliquent directement la convergence dans le sens Gromov–Hausdorff. Pour obtenir la convergence pour la topologie Gromov–Hausdorff–Prokhorov, nous prouvons que l’effet de l’attachement des masses sur l’objet limite est déterministe, si les masses sont attachées de manière échangeable et les masses sont uniformément asymptotiquement négligeables.

On Site Percolation in Random Quadrangulations of the Half-Plane

We study site percolation on uniform quadrangulations of the upper half plane. The main contribution is a method for applying Angel’s peeling process, in particular for analyzing an evolving boundary condition during the peeling. Our method lets us obtain rigorous and explicit upper and lower bounds on the percolation threshold \(p_\mathrm {c}\), and thus show in particular that \(0.5511\le p_\mathrm {c}\le 0.5581\). The method can be extended to site percolation on other half-planar maps with the domain Markov property.

The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection

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

The Brownian map is the scaling limit of uniform random plane quadrangulations

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 → ∞ 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.

The topology of scaling limits of positive genus random quadrangulations

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n\ge1$, a random quadrangulation $\mathfrak{q}_{n}$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph metric. As $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov–Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus $g$-torus.

Embedded trees and the support of the ISE

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by ±1. Recently it has been proved (see Chassaing and Schaeffer (2004) [8] and Janson and Marckert (2005) [11]) that the distribution of the maximum and minimum labels are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is to make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier et al. (2003) [6] that is based on properties of Jacobi’s θ-functions. In particular, we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass ℘-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer (2004) [8]) with the help of exact counting generating functions.

The two uniform infinite quadrangulations of the plane have the same law

We prove that the uniform infinite random quadrangulations defined respectively by Chassaing-Durhuus and Krikun have the same distribution.

Scaling Limits for Random Quadrangulations of Positive Genus

Abstract. We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every positive integer $n$, a random quadrangulation $q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n$ to the power of $-1/4$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to $4$. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter.

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometryof triangles made of three geodesic paths joining them. We also study the geometry ofminimal separating loops, i.e. paths of minimal length among all closed paths passing byone of the three vertices and separating the two others in the quadrangulation. Weconcentrate on the universal scaling limit of large quadrangulations, also known as theBrownian map, where pairs of geodesic paths or minimal separating loops have commonparts of non-zero macroscopic length. This is the phenomenon of confluence, whichdistinguishes the geometry of random quadrangulations from that of smooth surfaces. Wecharacterize the universal probability distribution for the lengths of these commonparts.

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton–Watson trees. As a consequence, we find that the expected volume of the ball of radius r around a marked point in the limit random surface is Θ(r4).

Invariance principles for random bipartite planar maps

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.

A conditional limit theorem for tree-indexed random walk

We consider Galton–Watson trees associated with a critical offspring distribution and conditioned to have exactly n vertices. These trees are embedded in the real line by assigning spatial positions to the vertices, in such a way that the increments of the spatial positions along edges of the tree are independent variables distributed according to a symmetric probability distribution on the real line. We then condition on the event that all spatial positions are nonnegative. Under suitable assumptions on the offspring distribution and the spatial displacements, we prove that these conditioned spatial trees converge as n → ∞ , modulo an appropriate rescaling, towards the conditioned Brownian tree that was studied in previous work. Applications are given to asymptotics for random quadrangulations.

Random planar lattices and integrated superBrownian excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R−L of the support of the one-dimensional ISE, or precisely: More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.

Decorating random quadrangulations

On various regular lattices (simple cubic, body centred cubic, etc) decorating an edge with an Ising spin coupled by bonds of strength L to the original vertex spins and competing with a direct anti-ferromagnetic bond of strength αL can give rise to three transition temperatures for suitable α. The system passes through ferromagnetic, paramagnetic, anti-ferromagnetic and paramagnetic phases respectively as the temperature is increased. For the square lattice on the other hand, multiple decoration is required to see this effect. We note here that a single decoration suffices for the Ising model on planar random quadrangulations (coupled to 2D quantum gravity). Other random bipartite lattices such as the Penrose tiling are more akin to the regular square lattice and require multiple decoration to have any affect.