Interlacing and Scaling Exponents for the Geodesic Watermelon in Last Passage Percolation

Abstract

In a discrete planar last passage percolation (LPP), random values are assigned independently 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 attached to the vertices of the path. The weight of a collection of disjoint paths is the sum of its members’ weights. The notion of a geodesic, namely a path of maximum weight between two vertices, has a natural generalization concerning several disjoint paths. Indeed, a k-geodesic watermelon in $$[1,n]^2\cap {\mathbb {Z}}^2$$ is a collection of k disjoint upright paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have remained largely unexplored beyond the $$k=1$$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $$k=1$$ the exponents that govern fluctuation in weight and transversal distance are 1/3 and 2/3; which is to say, the weight of the geodesic on the route $$(1,1) \rightarrow (n,n)$$ typically fluctuates around a dominant linear growth of the form $$\mu n$$ by the order of $$n^{1/3}$$ ; and the maximum Euclidean distance of the geodesic from the diagonal typically has order $$n^{2/3}$$ . Assuming a strong but local form of convexity and one-point moderate deviation estimates for the geodesic weight profile—which are available in all known exactly solvable models—we establish that, typically, the k-geodesic watermelon’s weight falls below $$\mu n k$$ by order $$k^{5/3}n^{1/3}$$ , and its transversal fluctuation is of order $$k^{1/3}n^{2/3}$$ . Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes. These bounds improve on estimates obtained by applying tools from the theory of determinantal point processes available in the integrable setting.