半直线上的开放量子随机游动：Karlin-McGregor公式、路径计数和Foster定理
Open quantum random walks on the half-line: the Karlin-McGregor formula, path counting and Foster's Theorem
摘要：
本文考虑非负整数上的开量子随机游动。通过考虑正交矩阵多项式，我们能够通过矩阵版本的Karlin-McGregor公式来描述一类行走的转移概率表达式。我们专注于吸收边界条件，对于简单的例子，我们考虑路径计数和相应的组合工具。研究了赌徒破产的一个非交换形式，得到了赌徒达到某一财富的概率和达到财富或破产的平均时间的生成函数。在阿达玛硬币的情况下，在一个格中的边界限制路径的计数技术。我们讨论了Foster定理的一个开放的量子版本的期望返回时间及其应用。
英文摘要：
In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.
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