Discrete holomorphicity

Project level: PhD, Masters

Lattice models are key tools in the analysis of a large class of physical systems in statistical mechanics. The inessential artefacts of the lattice are washed out in the continuum scaling limit. In this limit, many so-called critical lattice models in two dimensions are widely believed to be conformally invariant and admit holomorphic observables. However, this has only been established rigorously in very few cases. A key ingredient in these proofs is the introduction of lattice observables satisfying a discrete form of holomorphicity. This project aims to explore and extend recent breakthroughs on these matters. In a variety of lattice models, it will be examined how discrete complex analysis can be used to understand the emergence of holomorphic observables and how the existence of discrete holomorphicity is related to the notion of Yang-Baxter integrability of the lattice models.