School of Mathematics and Statistics - Theses
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ItemDiscretely holomorphic observables in statistical mechanics( 2014)In this thesis we investigate applications of discretely holomorphic observables in two-dimensional statistical mechanical lattice models. Discretely holomorphic observables or parafermionic ob- servables are functions defined on a graph embedding which depend on a real-valued parameter known as the parafermionic spin. Such observables were introduced in the context of lattice models in order to rigorously prove that the scaling limit of a given model is conformally invariant. This approach has been successfully carried out for the dimer model, percolation and the Ising model. However, these observables also have important applications at the lattice level. For example, Cardy and Ikhlef showed that they are naturally related to the Yang-Baxter integrability of the model, providing a straightforward method for obtaining the integrable weights. We further this connection by defining discretely holomorphic observables for loop models in domains with a boundary, showing that for a simple set of boundary conditions, the integrable boundary weights are obtained. This is true for models with diagonal as well as off-diagonal boundary weights, in which case the observables of Cardy and Ikhlef must be further generalised. Duminil-Copin and Smirnov made use of the discrete holomorphicity property in order to prove Nienhuis’ conjectured value for the connective constant of self-avoiding walks on the honeycomb lattice. We show that by relaxing the discrete holomorphicity condition we obtain an off-critical condition which allows us to relate certain critical exponents of different classes of self-avoiding walks in the dilute O(n) model. We are also able to derive an exponent equality and several exponent inequalities related to the winding angle distribution of the O(n) model, whose full distribution was first predicted using Conformal Field Theory techniques by Duplantier and Saleur. Finally we define parafermionic observables for the Andrews-Baxter-Forrester heights mod- els at the critical point. These models remain integrable away from the critical point. We therefore expect that the off-critical integrable weights should arise from an off-critical discrete holomorphicity condition, similar to that which we defined in the context of the dilute O(n) model.