正方形格子上简单走道的缠绕
Winding of simple walks on the square lattice
摘要：
描述了一种计算$\mathbb{Z}^2$上简单对角行走的方法，该方法的起点和终点固定在一个轴上，绕原点的缠绕角固定。该方法涉及到这样的散步分解成更小的块，其中的生成函数被编码在一个交换集的希尔伯特空间运营商。一般的计数问题，然后通过获得一个明确的特征值分解这些运营商涉及椭圆函数解决。通过进一步将走线的中间缠绕角限制到某个开区间，该方法可用于对限制到开度角为$π/4$的整数倍的$\mathbb{Z}^2$中的锥体的各种类别的走线进行计数。我们提出了三个应用程序的主要结果。首先，我们找到一个明确的生成函数的行走在这样的圆锥，开始和结束在原点。在特定的情况下，锥角$3π/4$这些步行直接相关的Gessel的步行在象限，我们提供了一个新的证明他们的枚举。接下来，我们研究的缠绕角分布的一个简单的随机游动在$\mathbb{Z}^2$周围的一个点在其起始点附近，我们确定离散类似物的已知双曲正割定律和概率解释的雅可比椭圆函数。最后，我们将希尔伯特空间中的一个算子的谱与$\mathbb{Z}^2$中围绕原点具有固定卷绕数的闭环的计数联系起来。
英文摘要：
A method is described to count simple diagonal walks on $\mathbb{Z}^2$ with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be used to count various classes of walks restricted to cones in $\mathbb{Z}^2$ of opening angles that are integer multiples of $π/4$. We present three applications of this main result. First we find an explicit generating function for the walks in such cones that start and end at the origin. In the particular case of a cone of angle $3π/4$ these walks are directly related to Gessel's walks in the quadrant, and we provide a new proof of their enumeration. Next we study the distribution of the winding angle of a simple random walk on $\mathbb{Z}^2$ around a point in the close vicinity of its starting point, for which we identify discrete analogues of the known hyperbolic secant laws and a probabilistic interpretation of the Jacobi elliptic functions. Finally we relate the spectrum of one of the Hilbert space operators to the enumeration of closed loops in $\mathbb{Z}^2$ with fixed winding number around the origin.
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