Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere

Recurrence of bipartite planar maps

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that for certain choices of weights the local limit has exactly one face of infinite degree and has in that case spectral dimension $4/3$ (the latter requires a mild moment condition).

On the sphericity of scaling limits of random planar quadrangulations

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.

Asymptotics for rooted planar maps and scaling limits of two-type spatial trees

We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when $n$ goes to infinity, a random $2\ka$-angulation with $n$ faces has a separating vertex whose removal disconnects the map into two components each with s…

Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

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…

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we class…

Geodesics in large planar maps and in the Brownian map

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points …

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behaviour of uniform random maps with a prescribed face-degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution towards the Brownian map in the Gromov-Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random $q$-angulations where $q$ is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of ed…

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 d…

Compact Brownian surfaces I. Brownian disks

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family ($\mathrm{BD}_L$, $0 < L < \infty$) of random metric spaces homeomorphic to the closed unit disk of $\mathbb{R}^2$, the space $\mathrm{BD}_L$ being called the Brownian disk of perimeter $L$ and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where $L = 0$. Similar results are obtained for maps following a Boltzmann distri…

从平面地图的边界度量看双尺度SYK模型

Double-scaled SYK from boundary metrics of planar maps

本文研究一类具有特殊q变形面权重的二分平面地图模型，证明其边界度量枚举仅依赖于测地线弦图的交叉数，并与双尺度SYK模型的弦图表示一致。

Limits of the boundary of random planar maps

We discuss asymptotics for the boundary of critical Boltzmann 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 parameter $α\in (1,2)$. First, in the dense phase corresponding to $α\in(1,3/2)$, we prove that the scaling limit of the boundary is the random stable looptree with parameter $(α-1/2)^{-1}$. Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to $α\in(3/2,2)$, it h…

The Brownian plane

We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the Gromov-Hausdorff tangent cone in distribution of the Brownian map at its root vertex, and it also arises as the scaling limit of uniformly distributed (finite) planar quadrangulations with n faces when the scaling factor tends to 0 less fast than n^{-1/4}. We discuss various properties of the Brownian plane. In particular, we prove that the Brownian plane is homeomorphi…

Scaling Limit of Random Planar Quadrangulations with a Boundary

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(σ_n)$ of integers such that $σ_n/\sqrt{2n}$ tends to some $σ\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2σ_n$ half-edges on the boundary. For $σ\in (0,\infty)$, we view $q_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric spac…

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-Hau…

随机穿孔双曲面与布朗球面

Random punctured hyperbolic surfaces & the Brownian sphere

研究按Weil-Petersson测度采样的n+1个穿孔0类随机双曲面Sₙ，证明度量经n⁻¹/⁴缩放后收敛于布朗球面，未缩放时局部收敛于特定无限体积双曲面，通过平面树编码等技术实现。

布朗球体的拟对称刚性

Quasisymmetric rigidity of the Brownian sphere

本文证明了布朗球体（布朗映射）几乎必然没有非平凡的拟对称自同构，且两个独立的布朗球体几乎必然不是拟对称等价的，从而建立了其拟对称刚性。