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
本文研究对称马尔可夫多项模型中球在高尔顿板上的最可能箱子，发现当转向次数达对数量级时，最可能箱子从角点直接跳至中心，并给出二阶修正常数。
AI 中文摘要
我们观察一个球在具有任意方向数的高尔顿板上弹跳。在每个钉子上，它要么以某个固定概率保持其方向，要么随机转向其他方向之一。其箱子度量它每次走向各方向的次数，即单纯形上的一个点。当球很少转向时，最可能的箱子是角点，这些角点只有从不转向的球才能到达。我们询问单纯形每个面的最佳箱子何时变得与角点一样可能。在一阶近似下，每个面在同一时刻赶上。我们在二阶近似下打破这种平局，为每个面给出一个显式常数。因此，在长板上，当预期转向次数增长到长度的对数的任意固定倍数时，最可能的箱子会从角点直接跳到中心，仅发生一次跳跃。非均匀起点或弱外部场会改变这些常数，可能允许某个中间面获胜。
英文摘要
We observe a ball bouncing down a Galton board with any number of directions. At each peg it either keeps its direction with some fixed probability or randomly turns to one of the other directions. Its bin measures how often it went each way, a point of a simplex. When the ball rarely turns, the most likely bins are the corners, which only a ball that never turns can reach. We ask when the best bin of each face of the simplex becomes as likely as a corner. To first order every face catches up at the same moment. We break this tie at second order, with an explicit constant for each face. Hence on a long board, as the expected number of turns grows to any fixed multiple of the length's logarithm, the most likely bin jumps once, from the corners straight to the center. A nonuniform start or a weak external field changes the constants, potentially allowing an intermediate face to win.
Comments39 pages, 2 figures. Companion to arXiv:2610.06935. Code and Lean files at https://github.com/apemm/Kagey-Problems
