Exploration of the arctic curve phenomenon in lattice statistical models via the tangent method - Debin, Bryan (2021)
One of the goals of statistical physics is to describe phase transitions occurring as some external parameters are varied. In this dissertation, we study what could be regarded as a spatial phase transition that manifests itself in various two-dimensional statistical models with specific boundary conditions. Typical equilibrium configurations of such models present two coexisting phases that are spatially separated by a sharp interface. In the scaling limit, this interface almost surely takes a fixed shape, which is called the arctic curve. The principal goal of this thesis is to apply and study the tangent method, a technique that was recently developed to determine arctic curves. We present the derivation of the arctic curve in various tiling and vertex models, using the tangent method. The core of the computation is the evaluation of the asymptotics of the partition function of the model under consideration with slightly modified (or refined) boundary conditions. The tangent method relies on two assumptions that we study in a variational framework. We provide strong arguments supporting the tangency property and formulate the factorization conjecture, which relates the asymptotic of (multi-)refined partition functions of a model to the shape of its arctic curve. This conjecture is shown to hold in full generality for the domino tilings of the Aztec diamond using exact lattice results. The factorization conjecture is also used to reformulate the tangent method. A numerical investigation of the fluctuations around a point of the arctic curve for Aztec diamonds exhibits distributions, originally found in random matrix theory, such as the Tracy-Widom distribution. (SC - Sciences) -- UCL, 2021