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In discrete planar last passage percolation (LPP), random values are assignedindependently to each vertex in $\mathbb Z^2$, and each finite upright path in$\mathbb Z^2$ is ascribed the weight given by the sum of values of itsvertices. The weight of a collection of disjoint paths is the sum of itsmembers' weights. The notion of a geodesic, a maximum weight path between twovertices, has a natural generalization concerning several disjoint paths: a$k$-geodesic watermelon in $[1,n]^2\cap\mathbb Z^2$ is a collection of $k$disjoint paths contained in this square that has maximum weight among all suchcollections. While the weights of such collections are known to be importantobjects, the maximizing paths have been largely unexplored beyond the $k=1$case. For exactly solvable models, such as exponential and geometric LPP, it iswell known that for $k=1$ the exponents that govern fluctuation in weight andtransversal distance are $1/3$ and $2/3$; that is, typically, the weight of thegeodesic on the route $(1,1) \to (n,n)$ fluctuates around a dominant lineargrowth of the form $\mu n$ by the order of $n^{1/3}$; and the maximum Euclideandistance of the geodesic from the diagonal has order $n^{2/3}$. Assuming astrong but local form of convexity and one-point moderate deviation bounds forthe geodesic weight profile---which are available in all known exactly solvablemodels---we establish that, typically, the $k$-geodesic watermelon's weightfalls below $\mu nk$ by order $k^{5/3}n^{1/3}$, and its transversal fluctuationis of order $k^{1/3}n^{2/3}$. Our arguments crucially rely on, and develop, aremarkable deterministic interlacing property that the watermelons admit. Ourmethods also yield sharp rigidity estimates for naturally associated pointprocesses, which improve on estimates obtained via tools from the theory ofdeterminantal point processes available in the integrable setting.

Interlacing and scaling exponents for the geodesic watermelon in last passage percolation