一个依赖空间的碎裂过程
A spatially-dependent fragmentation process
摘要：
我们定义了一个空间依赖的碎片化过程，它涉及到矩形打破成越来越小的块，取决于它们的形状的速率。长而细的矩形更容易快速断裂，也更容易沿着最长的边分裂。我们感兴趣的是系统如何随着时间的推移而演变：有多少不同形状和大小的碎片，它们是如何达到这种状态的？我们的定理给出了沿着路径的几乎必然增长率，这与预期的增长率不匹配--在某些路径上，具有这种形状和大小的碎片的预期数量是指数级的，但在现实中，几乎必然不存在这样的碎片。
英文摘要：
We define a spatially-dependent fragmentation process, which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Our theorem gives an almost sure growth rate along paths, which does not match the growth rate in expectation - there are paths where the expected number of fragments of that shape and size is exponentially large, but in reality no such fragments exist at large times almost surely.
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。
