Classification of scaling limits of uniform quadrangulations with a boundary

Abstract

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...

The incipient infinite cluster of the uniform infinite half-planar triangulation

“…For every n ∈ N, let M n be the map revealed at step n of the peeling process, and denote by M ∞ the underlying UIHPT (with origin vertex ρ). We denote by {τ • k : k ∈ Z + } and {τ • k : k ∈ Z + } the endpoints of the excursions intervals of B and W above their infimums, defined as in (7), and set φ(n)…”

“…Recall that by Tanaka's theorem, we have ξ N → ∞ P-a.s. as N → ∞. Let {τ k : k ∈ Z + }, {τ l k : k ∈ Z + } and {τ r k : k ∈ Z + } be the endpoints of the excursion intervals of A (r) , V (l) and V (r) above their infimum processes, as in (7). They define cut-points that disconnect the origin from infinity in L A (l) ,A (r) , L V (l) and L V (r) (and thus in H, H l and H r ) respectively, by the identities Since the processes A (r) , V (l) and V (r) are centered random walks and ξ N → ∞ P-a.s., we get K(N ), K l (N ), K r (N ) −→ N →∞ ∞ P-a.s..…”

Brownian geometry

“…One can prove versions of Theorem 9 for quadrangulations where both the boundary size and the volume (number of faces) are fixed and grow to infinity simultaneously in such a way that the volume stays proportional to the square of the boundary size: This leads to the definition of Brownian disks with given perimeter and volume. See [9] for a discussion of all possible scaling limits of quadrangulations with a boundary, and [33] for an analog of Theorem 9 in the case of quadrangulations with a simple boundary.…”

“…In particular, Theorem 19 relies on the identification of the distribution of the process giving, for each r ≥ 0, the sequence of boundary sizes of all excursions above level r of the process (Z a ) a∈T ζ under N * ,z 0 . There is a striking analogy with the fragmentation process occuring when cutting the CRT at a fixed height: Precisely, it is shown in [10] that the sequence of volumes of the connected components of the complement of the ball of radius r centered at the root in the CRT is a selfsimilar fragmentation process whose dislocation measure has the form (2π) −1/2 (x(1−x)) −3/2 dx, to be compared with the measure (x(1 − x)) −5/2 dx appearing in formula (9).…”

“…The Brownian half-plane, which also appears as the scaling limit of quadrangulations with a boundary when the volume and the boundary size tend to infinity in a suitable way, is discussed in [32] and [9] -a presumably equivalent construction had been given earlier by Caraceni and Curien [22]. The paper [9] provides an exhaustive study of possible scaling limits of quadrangulations with a boundary, leading to new models of Brownian geometry in addition to the Brownian map, the Brownian plane or the Brownian disk. In a series of recent papers, Miller and Sheffield [56,57,58,59] have developed a completely new approach to the Brownian map, showing also that this random compact metric space can be equipped with a conformal structure which is linked to Liouville quantum gravity.…”