We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α ∈ (1, 2). First, in the dense phase corresponding to α ∈ (1, 3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α − 1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α ∈ (3/2, 2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien & Kortchemski.

International audienceWe study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy’s formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map

We are interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α = 1. In stark contrast to the case α ∈ (1, 2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter 3/2). This supports the conjecture that faces in Le Gall & Miermont's 3/2-stable maps are self-avoiding.2010 Mathematics Subject Classification. -Primary 60J80 · 60G50 · 60F17 · 05C05; Secondary 05C80 · 60C05.

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