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We consider bipolar oriented random planar maps with heavy-tailed facedegrees. We show for each $\alpha \in (1,2)$ that if the face degree is in thedomain of attraction of an $\alpha$-stable L\'evy process, the correspondingrandom planar map has an infinite volume limit in the Benjamini-Schrammtopology. We also show in the limit that the properly rescaled contourfunctions associated with the northwest and southeast trees converge in law toa certain correlated pair of $\alpha$-stable L\'evy processes. Combined withother work, this allows us to identify the scaling limit of the planar map withan SLE$_\kappa(\rho)$ process with $\rho = \kappa-4 < -2$ on$\sqrt{\kappa}$-Liouville quantum gravity for $\kappa \in (4/3,2)$ where$\alpha, \kappa$ are related by $\alpha = 4/\kappa-1$.

Bipolar oriented random planar maps with large faces and exotic SLE$_\kappa(\rho)$ processes