We characterize all translation invariant half planar maps satisfying a certain natural domain Markov property. For p-angulations with p ≥ 3 where all faces are simple, we show that these form a one-parameter family of measures H (p) α . For triangulations we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well-known half plane uniform infinite planar maps. The sub-critical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.
We study non-compact 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 infinite 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 θ and the infinite-volume Brownian disk of perimeter σ. We also obtain various coupling and limit results clarifying the relation between these objects. (GRAAL) and ANR-15-CE40-0013 (Liouville), and of Fondation Simone et Cino Del Duca.Definition 2.1. Let T > 0. The continuum random tree CRT T with volume T is the random rooted real tree (T X , d x , ρ) for the probability distribution that makes the canonical process X of C([0, T ], R) the standard Brownian excursion with duration T .The term "CRT" usually denotes CRT 1 with volume T = 1, in which case X is taken under the law of the normalized Brownian excursion. We simply write CRT instead of CRT 1 .Note the scaling relation, for λ, T > 0:This comes from the fact that, if e T is a Brownian excursion with duration T , then λe T (·/λ 2 ) has same distribution as e λ 2 T .We should also discuss the role of ρ in the above definition. The re-rooting property of CRT T [2, (20)] states, roughly speaking, that if ρ is a random variable with distribution µ X /µ X (1) (the normalized version of the measure defined above), then (T X , d X , ρ ) has same distribution as (T X , d X , ρ). In this sense, the point ρ plays no distinguished role in the construction of CRT T .Brownian map BM T , T > 0. The Brownian map is roughly speaking the metric gluing of the tree coded by a snake driven by a normalized Brownian excursion, on the tree coded by the excursion itself.Definition 2.2. The Brownian map BM T with volume T is the metric space (M X,W , D X,W , ρ) for the probability law that makes X a Brownian excursion of duration T , and W is the snake driven by X.We write BM instead of BM 1 . The scaling properties of Gaussian processes imply easily that for λ > 0, λ · BM T = d BM λ 4 T .Just as for CRT T , the point ρ in BM T should be seen as a random choice according to the normalized measure µ X,W /µ X,W (1), which is known as the re-rooting property of the Brownian map (Theorem 8.1 of [28]). The latter is a crucial property for characterizing the Brownian map, see, e.g., the recent work [35].Brownian disk BD T,σ , σ ∈ (0, ∞), T > 0. The Brownian disk first appears in [7] as limiting metric space along suitable infinite subsequences. Uniqueness of the limit and a concrete description of the metric were obtained in [9].The description is slightly more elaborate than that of the Brownian map. For t ≥ 0, we let X t = inf [0,t] X.Definition 2.3. The Brownian disk BD T,σ with volume T and boundary length σ is the metric space (M X,W , D X,W , ρ) under the probability measures that makes X a f...
We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations.The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.
We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.
We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption. c pD 1 q has law Γ D 1 and ph D 1 D , φq " 0 a.s. for any φ with Supportpφq Ă DzD 1 .Remark 1.2. Note that in the domain Markov property, we have (by linearity) that if D 1 Ă D is simply connected, and φ 1 " φ 2 on D 1 , then ph D 1 D , φ 1 q " ph D 1 D , φ 2 q almost surely. When we discuss the domain Markov property later in the paper, we will often simply say that
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.