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Solving marginals of the LDP for the directed landscape
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We prove the upper-tail Large Deviation Principle (LDP) for the parabolic Airy process and characterize the limit shape of the directed landscape under the upper-tail conditioning. The LDP result answers Conjecture 10.1 in Das, Dauvergne, and Vir\'{a}g (2024). The starting point of our proof is the metric-level LDP for the directed landscape from Das, Dauvergne, and Vir\'{a}g (2024) that reduces our work to solving a variational problem. Our proof is PDE-based and uses geometric arguments, connecting the variational problem to the weak solutions of Burgers' equation. Further, our method may generalize to the setting of the upper-tail LDP for the KPZ fixed point under the multi-wedge initial data, and we prove a decomposition result in this direction.
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Cited by 1 Pith paper
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An upper tail field of the KPZ fixed point
A newly constructed 'upper tail field' is the local limit of the KPZ fixed point near a conditioned large value, interpolating between Brownian and KPZ scaling regimes.
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