Seed Seminar of Mathematics and Physics
# Statistical mechanics models on random maps of the disk and the half-plane

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**Abstract:**

In the main part of this 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 the Ising model with Dobrushin boundary conditions and at a fixed temperature (and without external magnetic field). We identify rigorously a phase transition by analysing the critical behaviour of the partition functions of a large disk at and around the critical temperature. Moreover, we study the random geometric implications of this in particular in a 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, which have interpretations in the continuum Liouville Quantum Gravity. The two key techniques in use are singularity analysis of rational parametrizations of generating functions, as well as the aforementioned exploration process. At the end of the talk, we will briefly discuss how this approach could be applied in further investigations of the Ising model and generalized to other statistical mechanics models on similar random lattices.