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Spacetime limit shapes of the KPZ equation in the upper tails

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arxiv 2304.14380 v1 pith:3TLOR2KF submitted 2023-04-27 math.PR math-phmath.MP

Yier Lin , Li-Cheng Tsai This is my paper · ORCID
classification math.PRmath-phmath.MP
keywords equationproofspacetimecorridorsdeviationslargelimitpoint
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abstract

We consider the $n$-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the $n$-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results -- from the companion paper Tsai (2023) -- on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.

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  1. An upper tail field of the KPZ fixed point

    math.PR 2025-01 conditional novelty 8.0 of 10

    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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