Statistical Models on Random Regular Graphs

Explore statistical models on random regular graphs in this 47-minute lecture by Alexander Gorsky from the Institut des Hautes Etudes Scientifiques (IHES). Delve into the application of the matrix-forest theorem and Parisi-Sourlas trick to formulate and solve a one-matrix model with non-polynomial potential. Discover how this model provides perturbation theory for massive spinless fermions on dynamical planar graphs, representing a discretized version of 2D quantum gravity coupled to massive spinless fermions. Learn about the model's equivalent description of spanning forest ensembles on the same graph, with solutions formulated using elliptic curves. Examine the near-critical scaling limit where both graphs and forest trees become macroscopically large, revealing universal one-point scaling functions parameterized by the Lambert function. Gain insights into the rare opportunity to explore the flow between two gravity models, specifically theories of conformal matter coupled to 2D gravity with c=-2 (large trees regime) and c=0 (small trees regime). Conclude with an overview of numerical simulation results concerning phase transitions in the Random Regular Graph (RRG) ensemble and their connection to Anderson localization.