Exact Exponents for Directed Distances in Planar Maps in the λγλ-LQG Universality Class

Explore exact scaling exponents for directed graph distances in random bipolar-oriented planar maps within the γ-Liouville quantum gravity universality class through this 45-minute conference talk. Discover how these maps, parameterized by γ∈(0,√(4/3)], exhibit distinct scaling behaviors where longest directed paths scale as n^(2/(4-γ²)) and shortest directed paths scale as n^(2/(4+γ²)) for maps of size n. Learn about the construction and analysis of the γ-UIBOT (the local limit around typical edges) and understand how the Busemann function, which measures directed distances to infinity along natural interfaces, converges to stable Lévy processes in scaling limits. Examine the mathematical framework connecting discrete map structures to conjectural γ-directed LQG metrics, including the convergence of Busemann functions to (1-γ²/4)-stable and (1+γ²/4)-stable Lévy processes for longest and shortest paths respectively. Gain insights into cutting-edge research at the intersection of probability theory, geometry, and quantum gravity through work conducted in collaboration with E. Gwynne, presented at IPAM's New Interactions Between Probability and Geometry Workshop.