Connection Probabilities for Loop O(n) Models and BPZ Equations

Explore the mathematical foundations of critical loop O(n) models and their conjectured conformal invariance in the scaling limit through this 33-minute conference talk. Delve into connection probabilities for loop O(n) models within polygonal domains, examining how these probabilities can be predicted using two distinct families of solutions to Belavin-Polyakov-Zamolodchikov (BPZ) equations: Coulomb gas integrals and SLE pure partition functions. Discover the theoretical framework connecting statistical mechanics models to conformal field theory, and learn about the proven cases where this conjecture holds true, including the critical Ising model, FK-Ising model, percolation, and uniform spanning tree. Gain insights into advanced probability theory and mathematical physics as the speaker presents rigorous mathematical analysis of these complex stochastic systems and their scaling behavior.