Permutation in Random Geometry

Random geometry and random permutations have been extremely active fields of research for several years. The former is characterized by studying large planar maps and their continuum limits, i.e., the Brownian map, Liouville quantum gravity surfaces, and Schramm–Loewner evolutions. The latter is characterized by studying large uniform permutations and (more recently) biased/pattern-avoiding permutations and their continuum limits, called permutons. These two fields had evolved completely separately until recently when some surprising connections emerged: it is possible to reconstruct some universal permutons directly using Liouville quantum gravity surfaces and Schramm–Loewner evolutions. We aim to report on these new connections that go through some naturally perturbed versions of the Tanaka stochastic equations.