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A number of interesting examples of random space filling curves have beenconstructed in two dimensional statistical mechanics models. Examples of theabove include SLE($8$), arising as the scaling limit of the Peano curve betweenthe uniform spanning tree of $\mathbb{Z}^2$ and its dual (Lawler-Schramm-Werner'04), or SLE($16/\gamma^2$), which describes the interface between two matedtrees in $\gamma$-Liouville Quantum gravity (Duplantier-Miller-Sheffield '14).In this work, we construct a Peano curve from the geodesic tree in a fixeddirection and its dual in the directed landscape, the putative universalspace-time scaling limit object in the KPZ universality class. The geodesictree and its dual were constructed in Bhatia '23, where it was shown that theyhave the same law up to reflection; indeed, the dual tree is the geodesic treeof the dual landscape as constructed there. We show that the interface betweenthe two trees is a space filling curve that is naturally parametrized by thearea and encodes the geometry of the two trees. We study the regularity andfractal properties of this Peano curve, exploiting simultaneously thesymmetries of the directed landscape and probabilistic estimates obtained inthe planar exponential last passage percolation, which is known to converge tothe directed landscape in the scaling limit.

A Peano curve from mated geodesic trees in the directed landscape