5 problems

For every integer $n\geq 2$, let $M_n$ be a uniformly distributed random rooted $2p$-angulation over ${\cal M}^p_n$, let ${\bf m}_n$ be its vertex set, and let $d_n$ be its graph d…

The infinite Brownian disk with boundary length $\mathcal{L}_1$ is a random metric space homeomorphic to the complement of the open unit disk in the plane. Shortest-loop conjecture…

Universality conjecture. The sequence of rescaled metric spaces should converge and, up to a deterministic multiplicative constant applied to the distance, the limit should be the…

Let ${\mathbf{c}_{\mathrm M}}\in(-\infty,25)$, and let $\mathcal S_h^\epsilon$ be the square subdivision graph associated with the field $h$, equipped with its graph distance. Dist…

Let $M_n$ be a uniform planar map with $n$ edges and let $Q_n$ be a uniform planar quadrangulation with $n$ faces. Equip both maps with their graph distances, and let the Ambjørn–B…