In this paper, we explain the connection between the elephant random walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced in [Phys. Rev. E 70, 045101 (2004)10.1103/PhysRevE.70.045101] to study memory effects in a highly non-Markovian setting. More specifically, the ERW is a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a memory parameter p between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of p. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.

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...

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We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition Π(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process Π is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of Π(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.