Related papers: Infinite random planar maps related to Cauchy proc…
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 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 start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a…
The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice…
The infinite discrete stable Boltzmann maps are generalisations of the well-known Uniform Infinite Planar Quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is…
We extend the peeling exploration introduced in arxiv:1506.01590 to the setting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its law is related to a class of discrete Markov processes obtained by confining random walks to…
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 $\alpha\in(1,2]$. We consider the critical Bernoulli bond…
We consider a model of a quenched disordered geometry in which a random metric is defined on ${\mathbb R}^2$, which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics,…
We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local…
Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a…
We discuss asymptotics for large random 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 index $\alpha\in(1,2)$. When the number $n$ of…
We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance…
The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The…
The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolation on maps with faces of arbitrary degree are…
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 $\alpha \in (1,2)$.…
We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as…
We study the phase diagram of random outerplanar maps sampled according to non-negative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its…
This thesis deals with some $(1+1)$-dimensional lattice path models from the KPZ universality class: the directed random polymer with inverse-gamma weights (known as log-gamma polymer) and its zero temperature degeneration, i.e. the last…
We are interested in the structure of large Bienaym\'e-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $\alpha=1$. In stark contrast to the case $\alpha…
We prove that large Boltzmann stable planar maps of index $\alpha \in (1;2)$ converge in the scaling limit towards a random compact metric space $\mathcal{S}_{\alpha}$ that we construct explicitly. They form a one-parameter family of random…