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 $\pi /4$.

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 $3\pi /4$ 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.

描述了一种计算简单的对角线行走的方法 ${\mathbb{ž}}^2$在其中一根轴上具有固定的起点和终点，并且绕原点具有固定的缠绕角。该方法涉及将这种走行分解为更小的片段，其生成函数被编码在希尔伯特空间算子的可交换集合中。然后，通过获得涉及椭圆函数的这些算子的显式特征值分解来解决一般枚举问题。通过将步道的中间缠绕角度进一步限制为某个开放间隔，该方法可用于计算限制在圆锥中的各种类型的步道。${\mathbb{ž}}^2$ 的开度角是的整数倍 $\pi /4$。

我们介绍了此主要结果的三种应用。首先，我们为在从原点开始和结束的圆锥体中的行走找到一个显式生成函数。在特定情况下为圆锥形$3\pi /4$这些走行与Gessel在象限中的走行直接相关，我们为它们的枚举提供了新的证明。接下来，我们研究简单随机游走的缠绕角度的分布${\mathbb{ž}}^2$围绕其起点附近的一点，我们确定了已知双曲正割定律的离散类似物以及Jacobi椭圆函数的概率解释。最后，我们将一个希尔伯特空间算子的频谱与闭环中的枚举相关联${\mathbb{ž}}^2$ 绕原点的绕组号固定。