Non-uniqueness times for the maximizer of the KPZ fixed point

Abstract

Let ${\mathfrak{h}}_t$ be the KPZ fixed point started from any initial condition that guarantees ${\mathfrak{h}}_t$ has a maximum at every time $t$ almost surely. For any fixed $t$, almost surely $\max {\mathfrak{h}}_t$ is uniquely attained. However, there are exceptional times $t\in (0,\infty )$ when $\max {\mathfrak{h}}_t$ is achieved at multiple points. Let ${\mathcal{T}}_k\subset (0,\infty )$ denote the set of times when $\max {\mathfrak{h}}_t$ is achieved at exactly $k$ points. We show that almost surely ${\mathcal{T}}_2$ has Hausdorff dimension $2/3$ and is dense, ${\mathcal{T}}_3$ has Hausdorff dimension $1/3$ and is dense, ${\mathcal{T}}_4$ has Hausdorff dimension $0$, and there are no times when $\max {\mathfrak{h}}_t$ is achieved at $5$ or more points. This resolves two conjectures of Corwin, Hammond, Hegde, and Matetski.

33 pages