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We develop a new probabilistic and geometric method to obtain several sharpresults pertaining to the upper tail behavior for the Kardar-Parisi-Zhang (KPZ)equation in a unified way. The arguments make crucial use of Brownianresampling invariance properties of Gibbs measures on infinite ensembles ofrandom continuous curves into which the KPZ equation (under narrow-wedgeinitial data) can be embedded. We obtain sharp one-point upper tail estimates,uniformly in the time parameter, showing that the probability of the value atzero being larger than $\theta$ is $\exp(-\frac{4}{3}\theta^{3/2}(1+o(1)))$,for a general class of initial data including the narrow-wedge. Our argumentsalso yield the first known density estimates for the narrow-wedge case, whichare optimal. A key intermediate step is developing a precise understanding ofthe profile when conditioned on the value at zero equaling $\theta$. Our methodfurther allows one to obtain multi-point asymptotics which were out of reach ofprevious approaches. As an example, we prove sharp explicit two-point uppertail estimates. All of our arguments also apply to the zero-temperature case ofthe Airy$_2$ process; even here, while some of the one-point estimates werealready available due to its connections to random matrix theory, the two-pointasymptotics are new. Finally, to showcase the reach of the method, we obtainthe same results in a purely non-integrable setting under only assumptions ofstationarity and extremality in the class of Gibbs measures, thus providingevidence in favour of a conjecture suggested by Sheffield and formulated byCorwin-Hammond on a characterization of such line ensembles. Our method bearsresemblance to the tangent method introduced by Colomo-Sportiello andmathematically realized by Aggarwal in the context of the six-vertex model.

Sharp upper tail estimates and limit shapes for the KPZ equation via the tangent method