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

Miermont, Grégory

doi:10.1007/s11511-013-0096-8

Cited by 277 publications

(289 citation statements)

References 34 publications

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“…The bijection between Hurwitz trees and increasing quadrangulations that we propose in the present paper can be seen as labelled counterparts to the bijections between plane trees and families of maps that are the basic building blocks of the approach that culminated with [13,15,14]. Hopefully they can lead to a proof of the above conjecture.…”

Section: B Large Random Increasing Quadrangulationsmentioning

confidence: 71%

“…In particular upon setting the edge length to n −1/4 , the random uniform quandrangulation Y n converges as a metric space to a continuum limit, the Brownian map, which is a random space with the topology of the sphere [13,15,14].…”

Section: B Large Random Increasing Quadrangulationsmentioning

confidence: 99%

“…Only later has it become possible to perform rigourous exact simulations via efficient (linear time) perfect random sampling [19,18,9]. The algorithmic technics underlying these samplers, mainly the identification of carefully chosen canonical spanning plane trees, have in turn triggered important progresses in the comprehension of the intrinsic geometries of random unlabelled quadrangulations [7], culminating with the construction of their continuum limit, the Brownian map [13,15,14]. We show here that a similar approach can be undertaken for simple branched coverings of the sphere: starting from a variant of the standard representation of factorizations as graphs embedded on surfaces, we first recast the problem in terms of some increasing quadrangulations.…”

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confidence: 99%

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Uniform random sampling of simple branched coverings of the sphere by itself

Duchi¹,

Poulalhon²,

Schaeffer³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abstract. We present the first polynomial uniform random sampling algorithm for simple branched coverings of the sphere by itself of degree n. More precisely, our algorithm generates in linear time increasing quadrangulations, which are equivalent combinatorial structures. Our result is based on the identification of some canonical labelled spanning trees, and yields a constructive proof of a celebrated formula of Hurwitz for the number of some factorizations of permutations in transpositions. The previous approaches were either non constructive or lead to exponential time algorithms for the sampling problem.Branched coverings of the sphere are 2-dimensional topological structures that have raised a lot of interest ever since the work of Hurwitz at the end of the 19th century. For instance, Okounkov and Pandharipande [17] have used these objects to derive an alternative to Kontsevitch's proof of Witten's celebrated conjecture. More recently, their relations to intersection numbers of moduli spaces and integrable hierachies as studied in mathematical physics have suggested that large random simple branched coverings provide an alternative model of discrete 2-dimensional pure quantum geometry (see e.g.[21] for a relatively accessible exposition). Our aim in the present article is to provide means to effectively sample these alternative random geometries, but since our approach is purely combinatorial we trade the topological definition of branched coverings for Hurwitz fundamental combinatorial representation (see however the appendix, and the complete and elegant treatment in [12]). Define a factorization in transpositions of the identity permutation id n on {1, . . . , n} to be a m-uple of transpositions τ 1 , . . . , τ m such that τ m · · · τ 1 = id n . It is transitive if the graph G τ on {1, . . . , n} with m edges given by the τ i is connected, and minimal if m = 2n − 2. It can be checked that indeed this is the minimum number of transpositions in a transitive factorization of id n .Theorem 1 (Hurwitz (1891)). Simple branched coverings of the sphere by itself of degree n are encoded up to homeomorphisms of the domain by minimal transitive factorizations in transpositions of the identity of S n , and their number, called n-th Hurwitz number, is n n−3 (2n − 2)!.The usual model of quantum geometries is the uniform distribution on fixed size unlabelled planar quadrangulations, which was first studied analytically [3] and via Markov chain simulations [2]. Only later has it become possible to perform rigourous exact simulations via efficient (linear time) perfect random sampling [19,18,9]. The algorithmic technics underlying these samplers, mainly the identification of carefully chosen canonical spanning plane trees, have in turn triggered important progresses in the comprehension of the intrinsic geometries of random unlabelled quadrangulations [7], culminating with the construction of their continuum limit, the Brownian map [13,15,14]. We show here that a similar approach can be undertaken for simple branched co...

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Section: B Large Random Increasing Quadrangulationsmentioning

confidence: 71%

Section: B Large Random Increasing Quadrangulationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Uniform random sampling of simple branched coverings of the sphere by itself

Duchi¹,

Poulalhon²,

Schaeffer³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Their scaling limits are also very different [LG13,Mie13]. We are not able to explain, not even at a heuristic level, why their Voronoi partitions would behave similarly.…”

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confidence: 62%

Voronoi tessellations in the CRT and continuum random maps of finite excess

Addario‐Berry¹,

Angel²,

Chapuy³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a large graph G and k agents on this graph, we consider the Voronoi tessellation induced by the graph distance. Each agent gets control of the portion of the graph that is closer to itself than to any other agent. We study the limit law of the vector Vor := (V 1 /n, V 2 /n, ..., V k /n), whose i'th coordinate records the fraction of vertices of G controlled by the i'th agent, as n tends to infinity. We show that if G is a uniform random tree, and the agents are placed uniformly at random, the limit law of Vor is uniform on the (k − 1)-dimensional simplex. In particular, when k = 2, the two agents each get a uniform random fraction of the territory. In fact, we prove the result directly on the Brownian continuum random tree (CRT), and we also prove the same result for a "higher genus" analogue of the CRT that we call the continuum random unicellular map, indexed by a genus parameter g ≥ 0. As a key step of independent interest, we study the case when G is a random planar embedded graph with a finite number of faces. The main idea of the proof is to show that Vor has the same distribution as another partition of mass Int := (I 1 /n, I 2 /n, ..., I k /n) where I j is the contour length separating the i-th agent from the next one in clockwise order around the graph.

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“…Are supercritical Galton Watson trees scale stationary? It follows from the work of Le Gall and Miermont, see [Gal11] and [Mie13], that the uniform infinite planar quadrangulation is a scale stationary distribution.…”

Section: Here Is Another Open Problemmentioning

confidence: 99%