Planar Quadrangulations Research Articles

Generating Plane Quadrangulations and Symmetry-preserving Operations on Maps

Lopsp-operations are operations on maps that are applied locally and are guaranteed to preserve all the orientation-preserving symmetries of maps. Well-known examples of such operations are dual, ambo, truncation, and leapfrog. They are described by plane 3-coloured triangulations with specific properties. We developed and implemented a program that can generate all lopsp-operations of a given size by reducing the problem of generating lopsp-operations to generating all plane quadrangulations that are not necessarily simple. We extended the program plantri to generate these quadrangulations.

Last car decomposition of planar maps

We give new equations which characterize the generating functions of planar quadrangulations and planar triangulations, with zero, one, or two boundaries. The proof is inspired by the Lackner–Panholzer last car decomposition of parking trees and consists in applying a similar decomposition to the peeling trees of planar maps.

Isoperimetric inequalities in the Brownian plane

We consider the model of the Brownian plane, which is a pointed noncompact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius r centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov property of the Brownian plane. Combining our study of short cycles with this strong spatial Markov property we obtain sharp isoperimetric bounds for the Brownian plane.

Conformal growth rates and spectral geometry on distributional limits of graphs

For a unimodular random graph (G,ρ), we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of (G,ρ), which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting consequences for many low-dimensional models. The consequences in dimension two are particularly strong. It establishes that models like the uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ) almost surely have spectral dimension at most two. It also establishes a conjecture of Benjamini and Schramm (Electron. J. Probab. 6 (2001) no. 23) by extending their recurrence theorem from planar graphs to arbitrary families of H-minor-free graphs. More generally, it strengthens the work of Gurel-Gurevich and Nachmias (Ann. of Math. (2) 177 (2013) 761–781) who established recurrence for distributional limits of planar graphs when the degree of the root has exponential tails. We further present a general method for proving subdiffusivity of the random walk on a large class of models, including UIPT and UIPQ, using only the volume growth profile of balls in the intrinsic metric.

Infinite stable Boltzmann planar maps are subdiffusive

The infinite discrete stable Boltzmann maps are generalisations of the well-known Uniform Infinite Planar Quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than 1/3. Our method is based on stationarity and geometric estimates obtained via the peeling process which are of own interest.

On Area-Universal Quadrangulations

We study drawings of plane quadrangulations such that every inner face realizes a prescribed area. A plane graph is area-universal if for every assignment of non-negative weights to the inner faces, there exists a straight-line drawing such that the area of each inner face equals the weight of the face. It has been conjectured that all plane quadrangulations are area-universal. We develop methods to prove area-universality via reduction to the area-universality of related graphs. This allows us to establish area-universality for large classes of plane quadrangulations. In particular, our methods are strong enough to prove area-universality of all plane quadrangulations with up to 13 vertices.

On the cyclic coloring conjecture

A cyclic coloring of a plane graph G is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph G is its cyclic chromatic number χc(G). Let Δ∗(G) be the maximum face degree of a graph G.In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected plane graph G has χc(G)≤⌊32Δ∗(G)⌋, it is enough to do it for subdivisions of simple 3-connected plane graphs.We have discovered four new different upper bounds on χc(G) for graphs G from this restricted family; three bounds of them are tight. As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple 3-connected plane quadrangulations, and simple 3-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple 3-connected plane graphs of maximum degree at least 10, and for subdivisions of simple 3-connected plane graphs having the maximum face degree large enough in comparison with the number of vertices of their longest paths consisting only of vertices of degree two.

Slit-Slide-Sew Bijections for Bipartite and Quasibipartite Plane Maps

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward the vertex. The faces of the new map have the same degree as those of the original map, except at the locations of the distinguished corners, where each receives an extra degree: this is the location of the distinguished half-edges. This bijection provides a sampling algorithm for uniform maps with prescribed face degrees and allows to recover Tutte's famous counting formula for bipartite and quasibipartite plane maps.
 In addition, we explain how to decompose the previous bijection into two more elementary ones, which each transfer a degree from one face of the map to another face. In particular, these transfer bijections are simpler to manipulate than the previous one and this point of view simplifies the proofs.

Markovian explorations of random planar maps are roundish

The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual.

A Markov Chain Sampler for Plane Curves

A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this article is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining this chain with flat histogram methods we achieve an efficient sampler of plane curves and knot diagrams. By analyzing data from this chain we are able to estimate the number of knot diagrams of a given size and also compute knotting probabilities and so investigate their asymptotic behavior.

Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation

We study geometric properties of the infinite random lattice called the uniform infinite planar quadrangulation or UIPQ. We establish a precise form of a conjecture of Krikun stating that the minimal size of a cycle that separates the ball of radius $R$ centered at the root vertex from infinity grows linearly in $R$. As a consequence, we derive certain isoperimetric bounds showing that the boundary size of any simply connected set $A$ consisting of a finite union of faces of the UIPQ and containing the root vertex is bounded below by a (random) constant times $|A|^{1/4}(\log|A|)^{-(3/4)-\delta}$, where the volume $|A|$ is the number of faces in $A$.

An improvement of Lebesgue’s description of edges in 3-polytopes and faces in plane quadrangulations

An edge e in a 3-polytope is of type (k1,k2,k3,k4) if the set of degrees of the vertices and faces incident with e is majorized by the vector (k1,k2,k3,k4).In 1940, Lebesgue proved that every 3-polytope has an edge of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5).This also provides a description of the faces of quadrangulated 3-polytopes in terms of degrees of their incident vertices.The purpose of our paper is to prove that every 3-polytope has an edge of one of the types (3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5), where all parameters except possibly 9 are best possible. We believe that 9 here is sharp and thus the whole description is tight.Our proof relies on the discharging method.

The three-state Potts antiferromagnet on plane quadrangulations

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations Γ. Any quadrangulation can be built from a dual pair (G,G*). Based on the duality properties of G, we propose a new criterion to predict the phase diagram of this model. If Γ is of self-dual type (i.e. if G is isomorphic to its dual G*), the model has a zero-temperature critical point with central charge c = 1, and it is disordered at all positive temperatures. If Γ is of non-self-dual type (i.e. if G is not isomorphic to G*), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge c = 4/5. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang–Swendsen–Kotecký Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type.

Statistics of the Voronoï cell perimeter in large bi-pointed maps

We study the statistics of the Voronoï cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance 2s and their Voronoï cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of s for maps with a large given volume N both in the scaling limit where s scales as N1/4, in which case the Voronoï cell perimeter scales as N1/2, and in the local limit where s remains finite, in which case the perimeter scales as s2 for large s. The obtained laws are universal and are characteristics of the Brownian map and the Brownian plane respectively.

Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations.

We provide a criterion based on graph duality to predict whether the three-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge c=1. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the three-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with very good agreement. We have also found that the Wang-Swendsen-Kotecký algorithm has no critical slowing-down in the former case, and critical slowing-down in the latter.

A universal law for Voronoï cell volumes in infinitely large maps

We discuss the volume of Voronoï cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations. We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoï cells keeps a finite volume, which scales as s4 for large s. We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.

On facial unique-maximum (edge-)coloring

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face α the maximal color appears exactly once on the vertices of α. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to 5. Fabrici and Göring (2016) even conjectured that 4 colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every 2-connected plane graph admits such a coloring with at most 4 colors.

Classification of scaling limits of uniform quadrangulations with a boundary

We study noncompact 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 self-similar 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.

On a conjecture by Chapuy about Voronoï cells in large maps

In a recent paper, Chapuy conjectured that, for any positive integer k, the law for the fractions of total area covered by the k Voronoï cells defined by k points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a uniform k-division of the unit interval. For , i.e. with two points chosen uniformly at random, it means that the law for the ratio of the area of one of the two Voronoï cells by the total area of the map is uniform between 0 and 1. Here, by a direct computation of the desired law, we show that this latter conjecture for actually holds in the case of large planar (genus 0) quadrangulations as well as for large general planar maps (i.e. maps whose faces have arbitrary degrees). This corroborates Chapuy’s conjecture in its simplest realizations.

Refined universal laws for hull volumes and perimeters in large planar maps

We consider ensembles of planar maps with two marked vertices at distance k from each other, and look at the closed line separating these vertices and lying at distance d from the first one (d < k). This line divides the map into two components, the hull at distance d which corresponds to the part of the map lying on the same side as the first vertex and its complementary. The number of faces within the hull is called the hull volume, and the length of the separating line the hull perimeter. We study the statistics of the hull volume and perimeter for arbitrary d and k in the limit of infinitely large planar quadrangulations, triangulations and Eulerian triangulations. We consider more precisely situations where both d and k become large with the ratio d/k remaining finite. For infinitely large maps, two regimes may be encountered: either the hull has a finite volume and its complementary is infinitely large, or the hull itself has an infinite volume and its complementary is of finite size. We compute the probability for the map to be in either regime as a function of d/k as well as a number of universal statistical laws for the hull perimeter and volume when maps are conditioned to be in one regime or the other.