## Pictures

- ©Loïc Richier

*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*.

- ©Loïc Richier

_{p},

*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 UIHPQ

_{p}admits a description in terms of a looptree, whose loops are filled in with finite-size quadrangulations with a simple boundary.

- ©Grégory Miermont

*n*inner faces and a boundary of size 2σ

_{n}, depending on the asymptotics of σ

_{n}and on the choice of the scaling factor

*a*

_{n}→ ∞. 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 IBD

_{σ}of perimeter

*σ*, the Brownian half-plane BHP, the Brownian half-plane BHP

_{θ}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 BD

_{σ}of perimeter

*σ*, the continuum random tree CRT, and the trivial one-point map.

- ©Igor Kortchemski

*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 X

_{n}of the walker at time

*n*in terms of a coarse-grained walker X̂ which takes steps of size

*L*; see the figure on the right.

_{n}- ©Wikipedia

*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.