Publications and preprints
- Condensation in critical Cauchy Bienaymé-Galton-Watson trees with Igor Kortchemski (preprint) (pdf)
ABSTRACT: 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.
- Duality of random planar maps via percolation with Nicolas Curien (preprint) (pdf)
ABSTRACT: We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter α∈(1,2]. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes α∈(3/2,2], and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime α∈(1,3/2), with parameter α′:=(2α+3)/(4α−2). This is the counterpart in random planar maps of the duality property κ↔16/κ of Schramm-Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.
- The boundary of random planar maps via looptrees with Igor Kortchemski (preprint) (pdf)
ABSTRACT: We study the scaling limits of the boundary of Boltzmann planar maps conditioned on having a large perimeter. We first deal with the non-generic critical regime, where the degree of a typical face falls within the domain of attraction of a stable law with parameter α∈(1,2). In the so-called dense phase α∈(1,3/2), it was established by Richier that the scaling limit of the boundary is a stable looptree. In this work, we complete the picture by proving that in the dilute phase α∈(3/2,2) (as well as in the generic critical regime), the scaling limit is a multiple of the unit circle. This establishes the first evidence of a phase transition for the topology of the boundary: in the dense phase, large faces are self-intersecting while in the dilute phase, they are self-avoiding. The subcritical regime is also investigated. In this case, we show that the scaling limit of the boundary is a multiple of the Brownian CRT instead. The strategy consists in studying scaling limits of looptrees associated with specific Bienaymé-Galton-Watson trees. In the first case, it relies on an invariance principle for random walks with negative drift, which is of independent interest. In the second case, we obtain the more general result that the Brownian CRT is the scaling limit of looptrees associated with BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution, confirming thereby a prediction of Curien & Kortchemski.
- The Incipient Infinite Cluster of the Uniform Infinite Half-Planar Triangulation (preprint) (pdf)
ABSTRACT: We introduce the Incipient Infinite Cluster (IIC) in the critical Bernoulli site perco- lation model on the Uniform Infinite Half-Planar Triangulation (UIHPT), which is the local limit of large random triangulations with a boundary. The IIC is defined from the UIHPT by conditioning the open percolation cluster of the origin to be infinite. We prove that the IIC can be obtained by adding within the UIHPT an infinite triangulation with a boundary whose distribution is explicit.
- Uniform infinite half-planar quadrangulations with skewness with Erich Baur (Electronic Journal of Probability Vol. 23, 2018) (pdf)
ABSTRACT: We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQp for short, with p ∈ [0,1/2] measuring the skewness). They interpolate between Kesten’s tree corresponding to p=0 and the usual UIHPQ with a general boundary corresponding to p=1/2. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that this family approximates the Brownian half-planes with skewness recently introduced by Baur, Miermont and Ray. For p<1/2, we give a description of the UIHPQp in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.
- Limits of the boundary of random planar maps (Probability Theory and Related Fields, online) (pdf)
ABSTRACT: 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 and Kortchemski.
- Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary with Erich Baur and Grégory Miermont (ALEA-Latin American Journal of Probability and Mathematical Statistics Vol. XIII, 2016) (pdf)
ABSTRACT: We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.
- Universal aspects of critical percolation on random half-planar maps (Electronic Journal of Probability Vol. 20, 2015) (pdf)
ABSTRACT: We 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.
ABSTRACT: This thesis deals with limits of large random planar maps with a boundary. First, we are interested in geometric properties of such maps. We prove scaling and local limit results for the boundary of Boltzmann maps whose perimeter goes to infinity, which we apply to the study of the rigid O(n) loop model on quadrangulations. Next, we introduce a family of random half-planar quadrangulations with a skewness parameter, and study their scaling limits and branching structure. Finally, we establish a confluence property of geodesics in uniform infinite half-planar maps, which are local limits of uniform triangulations and quadrangulations with a boundary. Second, we consider Bernoulli percolation models on uniform infinite half-planar maps. We compute the critical site percolation threshold for quadrangulations, and prove a universality property of these percolation models at criticality involving crossing probabilities. To conclude, we study the local limit of large critical percolation clusters by defining the incipient infinite cluster, a uniform infinite half-planar triangulation equipped with an infinite critical percolation cluster.