双向投票序列与随机游动的最大值分析
Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum
摘要：
考虑终止于其最大高度的非负晶格路径，这将被称为容许路径。我们表明，格路径可接受的概率与第一类或第二类切比雪夫多项式有关，具体取决于格路径是否定义有反射屏障。对给定长度的容许路径数或期望高度等参数进行了渐近分析。此外，我们还利用可容许随机游动与特殊二元序列之间的双射证明了最近由Zhao提出的关于选票序列的一个猜想。
英文摘要：
Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.
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