Jesper Jacobsen - Exact three- and four-point correlation functions in O(n) and Potts loop models

Recorded 22 May 2024. Jesper Jacobsen of the École Normale Supérieure presents "Exact three- and four-point correlation functions in the O(n) and Potts loop models" at IPAM's Vertex Models: Algebraic and Probabilistic Aspects of Universality Workshop.
Abstract: We give an overview of our recent work on geometrically defined bulk correlators in two-dimensional conformally invariant loop models. These correlators correspond to combinatorial maps, describing the connectivities between insertion points. Selected three-point correlators include the probability that three points belong to the same loop, or that two loops come close together in three points, or that an open curve running between two points pass through a third point. While the three-point correlators have a closed expression in terms of certain special functions, the four-point correlators are more involved and require deploying more technology. To determine them we combine the global symmetry of the CFT, the cellular algebra of its lattice discretisation, and interchiral conformal bootstrap. The 235 simplest four-point structure constants are found to be a product of a universal function of conformal dimensions, built from Barnes' double Gamma function, and a polynomial function of loop weights. The polynomial factors, which can be isolated by forming certain amplitude ratios, are retrieved in the lattice models and found to be independent of the size of the lattice, and even independent on whether the model stands at its critical point.
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