Statistical mechanics models on random lattices of the half-plane
Laboratoire de physique, ENS de Lyon
Lundi 13/06/2022, 11:00-12:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
In the main part of the talk, which is mostly based on joint works with Linxiao Chen, we start from a purely combinatorial problem of random planar triangulations of the disk coupled with an Ising model (either on the faces or the on the vertices) with Dobrushin boundary conditions and at a fixed temperature. We identify rigorously a phase transition by analysing the critical behaviour of the partition functions of a large disk at and around the critical point. Moreover, we study the random geometric implications of this in particular in the local limit when the disk perimeter tends to infinity. At the critical temperature, we also find some explicit scaling limits of observables related to the interface lengths as well as scaling limits of perimeter fluctuations associated with a Markovian exploration process of the half-plane Ising triangulation. The two key techniques in use are singularity analysis of rational parametrizations of generating functions, as well as the aforementioned exploration process.
In the remaining part (time permitting), I will explain more informally our ongoing project with Jérémie Bouttier and Grégory Miermont about how the above approach could be generalized to study random planar maps of a disk decorated with O(n) loop models (where rational parametrizations do not necessarily exist).