Abstracts > Da Silva William

Fortuin-Kasteleyn (FK) maps are a classical model of planar maps decorated with a percolation-like configuration, depending on a weight q > 0. For q < 4, Sheffield established that, in some appropriate sense, these maps converge after rescaling to a (decorated) Liouville quantum gravity surface with parameter depending on q. This scaling limit result builds on his celebrated hamburger-cheeseburger bijection, which provides a one-to-one correspondence between FK maps and a queueing model in a kitchen selling burgers. 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 this problem by showing that the walks can in fact be 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.