For general i.i.d. weights, the midpoint of the geodesic between (0,0) and (n,n) lying at position n/2+tn has probability e^{-2nJ_t(μ0)+o(n)}.
Upper tail large deviations of the directed landscape
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abstract
Starting from one-point tail bounds, we establish an upper tail large deviation principle for the directed landscape at the metric level. Metrics of finite rate are in one-to-one correspondence with measures supported on a set of countably many paths, and the rate function is given by a certain Kruzhkov entropy of these measures. As an application of our main result, we prove a large deviation principle for the directed geodesic.
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Large deviations of geodesic midpoint fluctuations in last-passage percolation with general i.i.d. weights
For general i.i.d. weights, the midpoint of the geodesic between (0,0) and (n,n) lying at position n/2+tn has probability e^{-2nJ_t(μ0)+o(n)}.