重建随机拼图
Reconstructing random jigsaws
摘要：
如果$n \times n$网格的边的着色由其$n^2$ \emph{tiles}的多集唯一地确定，则该着色被称为\emph{reconstructible}，其中对应于网格顶点的瓦片以某种固定顺序指定与该顶点关联的边的颜色。在2015年，Mossel和Ross提出了以下问题：如果$n \times n$网格的边缘使用$q=q(n)$不同的颜色随机地独立且均匀地着色，那么由此产生的着色是否可以以高概率重建？由下而上，Mossel和Ross证明了当$q = o(n^{2/3})$时，这样的着色是不可重建的;由上而下，Bordenave、Feige和Mossel以及Nenadov、Pfister和Steger分别证明了，对于任何固定的$ε> 0$，当$q \ge n^{1+ε}$时，这样的着色是可重建的。在这里，我们改进了这些结果，并证明了以下几点：存在绝对常数$C, c > 0$，使得作为$n \to \infty$，概率，一个随机着色如上所述是可重构的倾向于$1$，如果$q \ge Cn$和$0$，如果$q \le cn$。
英文摘要：
A colouring of the edges of an $n \times n$ grid is said to be \emph{reconstructible} if the colouring is uniquely determined by the multiset of its $n^2$ \emph{tiles}, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an $n \times n$ grid are coloured independently and uniformly at random using $q=q(n)$ different colours, then is the resulting colouring reconstructible with high probability? From below, Mossel and Ross showed that such a colouring is not reconstructible when $q = o(n^{2/3})$ and from above, Bordenave, Feige and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed $ε> 0$, that such a colouring is reconstructible when $q \ge n^{1+ε}$. Here, we improve on these results and prove the following: there exist absolute constants $C, c > 0$ such that, as $n \to \infty$, the probability that a random colouring as above is reconstructible tends to $1$ if $q \ge Cn$ and to $0$ if $q \le cn$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
