发表机构
Harvey Mudd College（哈维穆德学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究持久随机游走中端点穿越的均匀Bessel界，确定中间箱子与端部箱子等可能的条件，给出所有N的显式区间，并验证周期性边界的影响。
AI 中文摘要
一个球在高尔顿板上弹跳，其第一次弹跳的方向是公平的，而之后的每次弹跳以概率$p$重复前一次弹跳的方向，一直弹到某个箱子。我们询问对于哪些$p$，中间的箱子与端部箱子恰好等可能，这个问题最初由Kagey提出。我们发现，对于$N$次弹跳，当球平均改变方向（转弯）约$\log N$次时，会出现平局。我们将中心箱子与Bessel函数进行比较，其误差界对所有$N$都是显式的。这给出了每个$N \geq 2$时的平局的显式区间，而不仅仅是对于大的$N$。对于$N \geq 175$，这是通过解析方法完成的，对于较小的$N$，则通过整数算术精确完成，并在Lean中交叉验证。我们还发现了每个其他箱子与端部箱子的平局，以及周期性边界引起的偏移。
英文摘要
A ball bounces down a Galton board, where its first bounce's direction is fair while every later bounce repeats the previous bounce's direction with probability $p$, all the way down to some bin. We ask for which $p$ the middle bin is exactly as likely as the end bins, a question originally posed by Kagey. We find that for $N$ bounces, the tie occurs when the ball changes direction (turns) approximately $\log N$ times on average. We compare the central bin with a Bessel function, with an error bound that is explicit for all $N$. This yields an explicit interval for the tie at every $N \geq 2$, rather than just for large $N$. This is done analytically for $N \geq 175$, and for smaller $N$ exactly by integer arithmetic, cross-verified in Lean. We also find the tie of every other bin with the end bins and the shift caused by periodic boundaries.
Comments27 pages, 4 figures. Code, certificates and Lean proofs at https://github.com/apemm/Kagey-Problems (release paperA-arxiv-v1)
