Scaling limits of Fortuin-Kasteleyn planar maps at q=4

par William da Silva (Uni Wien)

jeudi 5 mars 2026, 14:00 → 15:00 Europe/Paris

435 (ENS)

In a landmark paper, Sheffield proposed a celebrated bijection encoding planar maps weighted by Fortuin-Kasteleyn percolation in terms of an inventory accumulation process in a kitchen producing hamburgers and cheeseburgers. His work initiated the Mating-Of-Trees approach to Liouville quantum gravity (LQG), as he identified the scaling limit of the hamburger-cheeseburger walks, which translates through the bijection as the convergence of Fortuin-Kasteleyn maps towards SLE/LQG in the so-called peanosphere sense. A striking feature of the result is a phase transition at the critical value q=4, where the limit degenerates. In this talk, we resolve the main open problem left by Sheffield's work by showing that the walks can be appropriately renormalised at criticality (q=4) provided we introduce a logarithmic correction that we identify exactly. The proof uses a novel strategy that reveals and makes use of the integrability of the model. Based on joint work with X. Hu, E. Powell and M. D. Wong.