一种随机拼图的散弹枪边缘组件
Shotgun edge assembly of random jigsaw puzzles
摘要：
在Mossel和Ross最近的工作中，有人问，对于一个由$q$不同形状的“jig”组成的随机拼图游戏，$q$必须有多大才能有一个解。在相同类型的两个夹具总是装配在一起的意义上，夹具被假定为对称的。他们表明，对于$q=o(n^{2/3})$，存在a.a.s.多种解决方案，对于$q=ω(n^2)$，有a.a.s.正好一个。后一个界限已经由Nenadov、Pfister和Steger以及Bordernave、Feige和Mossel独立地改进为$q\geq n^{1+\varepsilon}$。两组进一步指出，对于$q=o(n)$，存在a.a. s。拼图中重复的部分在本文中，我们表明，这样的难题a.a.s.只要$q\leq \frac{2}{\sqrt{e}}\,n - ω(\log_2 n)$，就有多个解，即使排列相同的片段不被认为会改变解。我们进一步给出一些意见的解决方案的数量，在这个政权的唯一解决方案的概率。
英文摘要：
In recent work by Mossel and Ross, it was asked how large $q$ has to be for a random jigsaw puzzle with $q$ different shapes of "jigs" to have exactly one solution. The jigs are assumed symmetric in the sense that two jigs of the same type always fit together. They showed that for $q=o(n^{2/3})$ there are a.a.s. multiple solutions, and for $q=ω(n^2)$ there is a.a.s. exactly one. The latter bound has since been improved to $q\geq n^{1+\varepsilon}$ independently by Nenadov, Pfister and Steger, and by Bordernave, Feige and Mossel. Both groups further remark that for $q=o(n)$ there are a.a.s. duplicate pieces in the puzzle. In this paper, we show that such puzzle a.a.s. has multiple solutions whenever $q\leq \frac{2}{\sqrt{e}}\,n - ω(\log_2 n)$, even if permuting identical pieces is not considered changing the solution. We further give some remarks about the number of solutions, and the probability of a unique solution in this regime.
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。
