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Does the most likely geodesic in last passage percolation with general weight distributions converge to a universal deterministic limiting shape under appropriate rescaling, and if so, what geometric properties characterize that shape?

Related: KPZ universality conjecture, Johansson's theorem on geodesic transversal fluctuations, Busemann function convergence in first passage percolation

Last passage percolation assigns random weights to points on a grid, and the geodesic is the path from corner to corner that maximizes the total collected weight. While the maximum value itself is well understood through KPZ universality theory, the actual path achieving that maximum, called the most likely or mode geodesic, is far less understood. The question is whether this distinguished path, conditioned on being the unique maximizer, converges after rescaling to some deterministic curve, and whether the shape of that curve is universal across different weight distributions or depends sensitively on the fine structure of those distributions.

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