Exceptional times when the KPZ fixed point violates Johansson's conjecture on maximizer uniqueness

In 2002, Johansson conjectured that the maximum of the Airy2 process minus the parabola x is almost surely achieved at a unique location [Joh03, Conjecture 1.5]. This result was proved a decade later by Corwin and Hammond [CH14, Theorem 4.3]; Moreno Flores, Quastel and Remenik [FQR13]; and Pimentel [Pim14]. Up to scaling, the Airy2 process minus the parabola x 2 arises as the fixed time spatial marginal of the KPZ fixed point when initialized from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when initialized from any element of a very broad class of initial data. None of these results rules out the possibility that at random times, the KPZ fixed point spatial marginal violates maximizer uniqueness. We prove that, for a very broad class of initial data, it is with positive probability that the set of such times is non-empty, and that, conditionally on this event, this set almost surely has Hausdorff dimension two-thirds. In terms of directed polymers, these times of maximizer non-uniqueness are instants of instability in the zero temperature polymer measure—moments at which the endpoint jumps from one location to another. Our analysis relies on the exact formula for the distribution function of the KPZ fixed point obtained by Matetski, Quastel and Remenik in [MQR21b], the variational formula for the KPZ fixed point involving the Airy sheet constructed by Dauvergne, Ortmann and Virág in [DOV18], and the Brownian Gibbs property for the Airy2 process minus the parabola x 2 demonstrated by Corwin and Hammond in [CH14].