Figure 1
©Loïc Richier
In a joint work with Grégory Miermont and Loïc Richier, we show that geodesic rays in the uniform infinite half-planar quadrangulation UIHPQ return to both sides of the boundary infinitely many times. The above figure presents an artistic drawing of the UIHPQ with two distinguished geodesics emanating from the root vertex ρ called the maximal or leftmost geodesic (in red) and the minimal or rightmost geodesic (in green). All geodesic rays starting from ρ lie in between the maximal and minimal geodesic. Their joint intersection points with the boundary are thus intersection points for any geodesic ray emanating from ρ.

Figure 2
©Loïc Richier
Together with Loïc Richier, we study a one-parameter family UIHPQp, p∈[0,1/2], of random quadrangulations of the half-plane, the uniform infinite half-planar quadrangulations with skewness. They interpolate between Kesten's tree (case p=0), that is, the local limit of a critical Galton-Watson trees conditioned to survive, and the standard UIHPQ with a non-simple boundary (case p=1/2). For 0<p<1/2, the UIHPQp admits a description in terms of a looptree, whose loops are filled in with finite-size quadrangulations with a simple boundary.

Figure 3
©Grégory Miermont
With Grégory Miermont and Gourab Ray, we classify the non-compact scaling limits of uniform random planar quadrangulations with n inner faces and a boundary of size 2σn , depending on the asymptotics of σn and on the choice of the scaling factor an → ∞. The following non-compact limiting spaces appear in distribution with respect to the local Gromov-Hausdorff topology (our spaces are rooted at the boundary): the Brownian plane BP, the infinite-volume Brownian disk IBDσ of perimeter σ, the Brownian half-plane BHP, the Brownian half-plane BHPθ with skewness θ, and the infinite continuum random tree ICRT. In a former work, Bettinelli showed that the compact scaling limits in the Gromov-Hausdorff topology are given by the Brownian map BM, the Brownian disk BDσ of perimeter σ, the continuum random tree CRT, and the trivial one-point map.

Figure 4
©Igor Kortchemski
In joint works with Jean Bertoin, we investigate various properties of large random recursive trees. A recursive or increasing tree on the integers {0,...,10} as shown on the left is a tree rooted at 0, such that the sequence of vertex labels along any branch from the root to another vertex increases. A random recursive tree of size 11 is a uniform choice of such an increasing tree on {0,...,10}. The figure on the right is a simulation of a random recursive tree of size 15000 made by Igor Kortchemski (the lengths of the edges vary to enable a planar representation). We study a dynamical version of a Bernoulli bond percolation on such trees, where edges are deleted one after the other in a random uniform order. Depending on how many edges are removed, different (percolation) regimes appear when the size of the tree tends to infinity.

Figure 5
In collaboration with Erwin Bolthausen, we study the spatial behavior of random walks in an i.i.d. random environment on the d-dimensional grid, d≥3, which is a small perturbation of the environment corresponding to simple random walk. Under a centering condition, we prove that the exit laws from large balls are close to those of a perturbed simple random walk. As indicated on the left, the idea is to transport suitable estimates on exit laws from balls on a smaller scale to the next largest scale, via an inductive scheme. In a separate work, we combine the spatial estimates with a control over sojourn times and prove diffusive behavior of random walks in random environment which are balanced in one fixed coordinate direction. This is achieved by expressing the position Xn of the walker at time n in terms of a coarse-grained walker X̂ which takes steps of size Ln ; see the figure on the right.

Figure 6
Together with Jean Bertoin, we demonstrate the connection of certain urn models à la Pólya to a model of a random walk with memory introduced by Schütz and Trimper in 2004 called the elephant random walk. The influence of the memory is measured by a parameter p taking values between 0 and 1. Using known results from the theory of urns, we obtain functional limit theorems for the elephant random walk, proving diffusive Gaussian behavior for p<3/4, marginally superdiffusive Gaussian behavior in the critical case p=3/4, and superdiffusive non-Gaussian behavior for p>3/4. The above figure depicts an elephant performing a non-random walk.