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

Abstract

We study the asymptotic behavior 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 toward 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 edges, vertices, or faces. The proof relies on the convergence of so-called "discrete snakes" obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees. K E Y W O R D SBrownian map; Brownian snake; labelled trees; limit theorems; random maps Random Struct Alg. 2018;00:1-56.wileyonlinelibrary.com/journal/rsadefines a probability measure on M. We consider next such random maps conditioned to have a large size for several notions of size. For every integer n ≥ 1, let M E=n , M V =n and M F=n be the subsets of M of those maps with respectively n edges, n vertices and n faces. For every S = {E, V , F} and every n ≥ 1, we definethe law of a Boltzmann map conditioned to have size n; here and later, we shall always, if necessary, implicitly restrict ourselves to those values of n for which W q (M S=n ) � = 0, and limits shall be understood along this subsequence. Under mild integrability conditions on q, we prove in Section 7 that for every S ∈ {E, V , F}, there exists a constant K q S > 0 such that if M n is sampled from P q S=n for every n ≥ 1, then the convergence in distributionholds in the sense of Gromov-Hausdorff. We refer to Theorem 3 for a precise statement. Observe that for any choice S ∈ {E, V , F}, if M n is sampled from P q S=n then, conditional on its degree sequence, say, ν Mn = (ν Mn (i); i ≥ 1), it has the uniform distribution in M(ν Mn ). The proof of the above convergence consists in showing that ν Mn satisfies (H) in probability for some deterministic limit law p q . Indeed, by Skorohod's representation Theorem, there exists then a probability space where versions of ν Mn under P q S=n satisfy (H) almost surely so we may apply Theorem 1 and conclude the convergence in law of the rescaled maps.The case S = V was obtained by Le Gall [27, Theorem 9.1], relying on results of Marckert and Miermont [33], when q is regular critical, meaning that the distribution p q (which is roughly that of the half-degree of a typical face when we see vertices as faces of degree 0) admits small exponential moments. Here, we generalize this result (and consider other conditionings) to all generic critical sequences q, that is, those for which p q admits a second moment.

Limits of the boundary of random planar maps

“…This result has been extended by Le Gall [45] to critical sequences q such that the degree of a typical face has exponential moments (while the first results on this model were obtained by Marckert & Miermont [49]). The result also holds for critical sequences q such that the degree of a typical face has a finite variance, as shown in the recent work [51] (such a sequence is called generic critical, see Section 2.1 for precise definitions). Convergence towards the Brownian map has also been established in the non-bipartite case in [52,54].…”

Condensation in critical Cauchy Bienaymé–Galton–Watson trees

“…After the seminal papers [MM06, LG07], Le Gall [LG13] and Miermont [Mie13] proved that uniform quadrangulations have a scaling limit, the Brownian map. This convergence was later extended to generic critical sequences in [Mar18b], building on the earlier works [MM07,LG13]. In 2011, Le Gall and Miermont [LGM11] established the subsequential convergence of non-generic critical Boltzmann maps.…”

Brownian disks and the Brownian snake

“…It was proved independently in [43] for quadrangulations and in [33] for more general cases including triangulations, that several important classes of random planar maps converge in this sense toward a limiting random compact metric space called the Brownian map, which had been introduced previously in [40]. Several recent papers (see in particular [1,4,9,41]) have shown that many other classes of random planar maps converge to the Brownian map, which thus provides a universal continuous model of random geometry in two dimensions. On the other hand, in a series of recent papers [45,46,47,48], Miller and Sheffield have shown that the Brownian map can be equipped with a conformal structure, which is linked to Liouville quantum gravity, and that this conformal structure is in fact determined by the Brownian map viewed as a random metric space.…”