We study a natural one-parameter family of random bipolar-oriented planar maps which lies in the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma\in(0, \sqrt{4/3}]$. 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 $n^{2/(4-\gamma^2)}$ (up to constants), while the shortest directed paths have lengths of order $n^{2/(4+\gamma^2)}$ (up to constants). These exponents give the scaling dimensions of discretizations of the conjectural $\gamma$-directed LQG metrics for $\gamma\in(0, \sqrt{4/3}]$. These results are obtained by analyzing the local limit, denoted by $\gamma$-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 $\gamma$-UIBOT. We show that in the case of longest (resp.\ shortest) directed paths, this Busemann function converges in the scaling limit to a $(1-\gamma^2/4)$-stable Lévy process (resp.\ a $(1+\gamma^2/4)$-stable Lévy process). Based on joint work with E. Gwynne.
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