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Explore directed distances in bipolar-oriented triangulations through this 47-minute conference talk that examines longest and shortest directed paths in uniform infinite bipolar-oriented triangulations (UIBOT). Learn about the construction of the Busemann function that measures directed distance to infinity along natural interfaces, and discover how this function converges in scaling limits to stable Lévy processes - specifically 2/3-stable for longest paths and 4/3-stable for shortest paths. Examine up-to-constants bounds for directed distances in finite bipolar-oriented triangulations from Boltzmann distributions and size-n cells in UIBOT, revealing that longest directed path lengths scale as n^(3/4) and shortest as n^(3/8) in typical n-edge subsets. Understand how these results provide scaling dimensions for discretizations of hypothetical √(4/3)-directed Liouville quantum gravity metrics, with potential applications to other random planar map models and longest increasing subsequences in pattern-avoiding permutations.
