On Ising Model Coupled to Random Planar Triangulations
Joonas Turunen · Työväentutkimus Vuosikirja · 2020
Random planar maps have been widely studied, due to their role as a discretization of the random surfaces in Liouville quantum gravity. However, there are still relatively few mathematical studies concerning random planar maps coupled to statistical physics lattice models, which in turn model Liouville quantum gravity coupled with matter fields. A canonical model of that kind is the Ising model, originally introduced to model ferromagnetic matter. This dissertation improves understanding of the geometry of random planar triangulations coupled to the Ising model at the critical temperature and away from it, starting from the discrete level, via an exploration process following the interface on an Ising-decorated random triangulation. The exploration is called the peeling process. The interface is imposed by boundary conditions consisting of two consecutive arcs of opposite spins, which is the most natural setting involving a single non-closed interface. In the first article of the thesis, it is shown that when the spins lie on the faces and the temperature is critical, there exists an infinite volume limit of the model, in the sense of local weak convergence, as the boundary size tends to infinity one arc after another. Then, the geometry of the interface is studied via certain observables derived from the peeling process. It is shown that the interface drifts almost surely towards infinity, and moreover some scaling limits related to the interface length are obtained. In the second article, the same model is studied outside the critical point, showing a phase transition predicted by the physics literature. This involves showing the local convergence of the model for any temperature and studying the geometry of the local limit in different temperatures. Moreover, the local convergence of the first article is generalized to the case where the length of the two boundary arcs tend to infinity simultaneously. The last article of the thesis concerns a similar model where the Ising spins lie on the vertices of the triangulation instead of the faces. This model has a different behaviour in the high-temperature regime, and otherwise exhibits universality with the model concerned in the first two articles. Moreover, the fact that the interface is a simple curve gives an explicit scaling limit of the interface length, revealing a connection to the continuum Liouville quantum gravity.