论文标题
KPZ固定点最大化器的非唯一时间
Non-uniqueness times for the maximizer of the KPZ fixed point
论文作者
论文摘要
让$ \ mathfrak h_t $是KPZ固定点，从保证$ \ Mathfrak H_T $的任何初始条件开始，几乎每次$ t $几乎肯定。对于任何固定的$ t $，几乎肯定$ \ max \ mathfrak h_t $都是独特的。但是，当$ \ max \ mathfrak h_t $在多个点上实现$ \ max \ mathfrak h_t $时，有非凡的时间$ t \ in（0，\ infty）$。令$ \ mathcal t_k \ subset（0，\ infty）$表示$ \ max \ maxfrak h_t $以$ k $点的实现。 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.这解决了Corwin，Hammond，Hegde和Matetski的两个猜想。
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.
