Non-existence of three non-coalescing infinite geodesics with the same direction in the directed landscape

It is believed that for metric-like models in the KPZ class the following property holds: with probability one, starting from any point, there are at most two semi-infinite geodesics with the same direction that do not coalesce. Until now, such a result was only proved for one model - exponential LPP (Coupier, D., in Electron. Commun. Probab. 16, 517–527, 2011) using its inherent connection to the totally asymmetric exclusion process. We prove that the above property holds for the directed landscape, the universal scaling limit of models in the KPZ class. Our proof reduces the problem to one on line ensembles and therefore paves the way to show similar results for other metric-like models in the KPZ class. Finally, combining our result with the ones in (Bhatia, M., Duality in the directed landscape and its applications to fractal geometry. https://arxiv.org/pdf/2301.07704 , 2023: Busani, O., et al. To appear in Annals of Probability, (2022)) we obtain the full qualitative geometric description of infinite geodesics in the directed landscape.