In 2002, Johansson conjectured that the maximum of the Airy process minus the parabola 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 Airy process minus the parabola arises as the fixed time spatial marginal of the KPZ fixed point when started from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when begun 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. To understand this possibility, we study the probability that the KPZ fixed point has, at a given time, two or more locations where its value is close to the maximum, obtaining quantitative upper and lower bounds in terms of the degree of closeness for a very broad class of initial data. We also compute a quantity akin to the joint density of the locations of two maximizers and the maximum value. As a consequence, the set of times of maximizer non-uniqueness almost surely has Hausdorff dimension at most two-thirds. Our analysis relies on the exact formula for the distribution function of the KPZ fixed point obtained by Matetski, Quastel and Remenik in [MQR21], the variational formula for the KPZ fixed point involving the Airy sheet constructed by Dauvergne, Ortmann and Virág in [DOV], and the Brownian Gibbs property for the Airy process minus the parabola demonstrated by Corwin and Hammond in [CH14].
I. Corwin was partially supported by the NSF grants DMS-1811143 and DMS-1664650, and a Packard Fellowship for Science and Engineering. A. Hammond was partially supported by NSF grant DMS-1512908 and a Miller Professorship from the Miller Institute for Basic Research in Science. M. Hegde was partially supported by the Richman Fellowship of the U.C. Berkeley mathematics department and NSF grant DMS-1855550. K. Matetski was partially supported by NSF grant DMS-1953859.
"Exceptional times when the KPZ fixed point violates Johansson’s conjecture on maximizer uniqueness." Electron. J. Probab. 28 1 - 81, 2023. https://doi.org/10.1214/22-EJP898