环上相互竞争的随机游走之间的界面
Interface between competing random walks on a cycle
浏览论文内容
中文总结 AI 辅助
该研究探讨环上两独立随机游走者的竞争，证明端点被不同游走者占据的边的期望数量为$\text{O}(\text{ln}(1+N/d))$，证实了相关论文预测的对数依赖关系。
中文摘要 AI 辅助
我们考虑长度为$N$的环上两个独立随机游走者之间的竞争：每个顶点会被首先到达该顶点的游走者占据，且之后保持被占据状态。我们证明，若两个游走者的初始距离为$d$，则端点被不同游走者占据的边的期望数量为$\boldsymbol{\text{O}}(\boldsymbol{\text{ln}}(1+N/d))$。这证实了Gomes Jr.等人在《Physica A: Statistical Mechanics and its Applications》1996年第225卷第1期第81-88页发表的论文《Coloring of a one-dimensional lattice by two independent random walkers》中预测的对$N/d$的对数依赖关系。
英文摘要
We consider a competition between two independent random walks on a cycle of length $N$. Each vertex is claimed by the walker that visits it first, and remains claimed thereafter. We prove that if the initial distance between the walkers is $d$, then the expected number of edges whose endpoints are claimed by different walkers is of order $\ln(1+N/d).$ This confirms the logarithmic dependence on $N/d$ predicted in Gomes Jr. et al. [Coloring of a one-dimensional lattice by two independent random walkers. Physica A: Statistical Mechanics and its Applications 225.1 (1996): 81-88].
