Recorded 29 January 2026. Jacopo Borga of the Massachusetts Institute of Technology presents "Exact exponents for directed distances in planar maps in the λ\gammaλ-LQG universality class for λ\gamma\in(0, \sqrt{4/3}]λ" at IPAM's New Interactions Between Probability and Geometry Workshop. Abstract: We study a natural one-parameter family of random bipolar-oriented planar maps which lies in the γ -Liouville quantum gravity (LQG) universality class for γ∈(0,4/3−−−√] . For these maps, we identify exact scaling exponents for directed graph distances. Writing n for the size of the map, the longest directed paths have lengths comparable to n2/(4−γ2) (up to constants), while the shortest directed paths have lengths of order n2/(4+γ2) (up to constants). These exponents give the scaling dimensions of discretizations of the conjectural γ -directed LQG metrics for γ∈(0,4/3−−−√] . These results are obtained by analyzing the local limit, denoted by γ -UIBOT, of these maps around a typical edge. We construct the Busemann function, which measures directed distances to infinity along a natural interface in the γ -UIBOT. We show that in the case of longest (resp.\ shortest) directed paths, this Busemann function converges in the scaling limit to a (1−γ2/4) -stable Lévy process (resp.\ a (1+γ2/4) -stable Lévy process). Based on joint work with E. Gwynne. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/new-interactions-between-probability-and-geometry/?tab=overview