Universality and Rotational Invariance in Isoradial Graphs

This lecture discusses the concepts of universality and rotational invariance in the context of isoradial graphs. The instructor presents theorems related to these concepts, starting with the theorem of rotational invariance, which states that any sub-sequential limit of a certain sequence is invariant under rotations. The discussion then transitions to the universality theorem, which asserts that within a class of models, the scaling limit remains consistent. The instructor elaborates on isoradial graphs, explaining their construction and significance in statistical mechanics, particularly in relation to FK percolation. The lecture highlights the star-triangle transformation, a key property of isoradial graphs that preserves connection laws. The instructor also addresses the implications of these transformations for the geometry of clusters formed in percolation models. The session concludes with insights into the convergence of Gaussian Free Fields (GFF) and the relationship between these mathematical structures and physical models, emphasizing the importance of understanding these concepts for further research in the field.