Geometry of large Boltzmann outerplanar maps

Abstract

We study the phase diagram of random outerplanar maps sampled according to nonnegative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The Boltzmann outerplanar maps are then shown to converge in the Gromov-Hausdorff sense towards the -stable looptree introduced by Curien and Kortchemski (2014), with the parameter depending on the specific weight-sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree. KEYWORDSenriched trees, looptrees, outerplanar maps, random trees INTRODUCTIONA planar map may be described as a proper embedding of a connected graph into the sphere, considered up to continuous orientation-preserving deformations. In order to eliminate possible internal symmetries one usually distinguishes and orients a root edge. The probabilistic study of these objects has lead to a rich and beautiful theory, see for example the survey paper [31] and references given therein. The connected components of the complement of a planar map are its faces, and the unique face that lies to the right of the oriented edge is termed the outer face. This face is usually drawn as the unbounded face in plane representations. The number of edges adjacent to a face is its degree. A planar map is termed outerplanar, if all its vertices are adjacent to the outer face. Classes of outerplanar maps have received some attention in recent literature: Bonichon, Gavoille, and Hanusse [5] gave a combinatorial encoding of outerplanar maps in terms of certain bi-colored plane trees. Combining this encoding with probabilistic techniques, Caraceni [8] described the asymptotic geometric shape of uniform outerplanar maps, establishing Aldous' Brownian tree as their Gromov-Hausdorff scaling limit. The scaling limit and the asymptotic enumerative formula for Random Struct Alg. 2019;55:742-771.wileyonlinelibrary.com/journal/rsa

The boundary of random planar maps via looptrees

“…Second, it is often challenging to relax an assumption involving a finite exponential moment condition to a finite variance condition: in particular, the proof of Theorem 1.2 uses different techniques than in [17,Theorem 13], and new ideas. We emphasize that until now, convergence towards the Brownian CRT of similar rescaled discrete weighted tree-like structures has mostly been obtained under finite exponential moment conditions (see [17,Theorems 1,13 and 14], [48,Theorem 5.1], [53,Theorem 6.60], and in particular the discussion in [51,Section 3.3]). Third, the method is robust, as it allows to treat the case σ 2 µ = ∞, which was left as an open question in [18].…”

Infinite stable looptrees

“…For example, the same authors showed in [CK15] that a stable looptree arises as the scaling limit of the boundary of a critical percolation cluster on the UIPT, and Richier showed in [Ric18a] that the incipient infinite cluster of the UIHPT has the form of an infinite discrete looptree. Further results along these lines can be found in [CK15], [CDKM15], [SS19], [BR18], [CR18] and [KR], though this is a very non-exhaustive list. More generally, they also arise as the scaling limits of boundaries of stable maps [Ric18b], and are emerging as an important tool in the programme to reconcile the theories of random planar maps and Liouville quantum gravity, demonstrated for example in [MS15], [GP] and [BHS18].…”

Asymptotic properties of random unlabelled block-weighted graphs

“…Second, further studies of different weight sequences may lead to the discovery of new phenomena. Contemporary examples include the α-stable maps by Le Gall and Miermont [38] and limit theorems for face-weighted outerplanar maps in [45].…”