Related papers: Bipolar oriented random planar maps with large fac…
The natural paramterization or length for the Schramm-Loewner evolution (SLE{\kappa}) is the candidate for the scaling limit of the length of discrete curves for \kappa < 8. We improve the proof of the existence of the parametrization and…
We discuss asymptotics of large Boltzmann random planar maps such that every vertex of degree $k$ has weight of order $k^{-2}$. Infinite maps of that kind were studied by Budd, Curien and Marzouk. These maps can be seen as the dual of the…
We consider several aspects of the scaling limit of percolation on random planar triangulations, both finite and infinite. The equivalents for random maps of Cardy's formula for the limit under scaling of various crossing probabilities are…
The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is…
A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series-parallel maps…
These are the notes for a series of lectures given at the Clay Mathematical Institute Summer School in Buzios, July 11 - August 7, 2010. We review some of the recent aspects of scaling limits of random trees and planar maps, in particular…
We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version…
We introduce a new family of random compact metric spaces $\mathcal{S}_\alpha$ for $\alpha\in(1,2)$, which we call stable shredded spheres. They are constructed from excursions of $\alpha$-stable L\'evy processes on $[0,1]$ possessing no…
We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations…
Various features of the two-parameter family of Schramm-Loewner evolutions SLE(\kappa,\rho) are studied. In particular, we derive certain restriction properties that lead to a ``strong duality'' conjecture, which is an identity in law…
We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab\'asi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& R\'acz…
SLE(kappa,rho) is a generalisation of Schramm-Loewner evolution which describes planar curves which are statistically self-similar but not conformally invariant in the strict sense. We show that, in the context of boundary conformal field…
These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of…
We study the geometry of infinite random Boltzmann planar maps with vertices of high degree. These correspond to the duals of the Boltzmann maps associated to a critical weight sequence $(q_{k})_{ k \geq 0}$ for the faces with polynomial…
We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_\kappa$) curve $\gamma$ for $\kappa\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where…
The scaling limit of planar loop-erased random walks is described by a stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the upper half-plane H minus a simply-connected compact subset K of H is studied. As a main…
We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic…
The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this…
We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…
We study SLE$_\kappa(\rho)$ curves, with $\kappa$ and $\rho$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure…