论文标题
计数格子绕角
Counting lattice walks by winding angle
论文作者
论文摘要
我们解决了通过缠绕角度在Kreweras Lattice上进行绕组的问题，Kreweras Lattice是三角形晶格的定向版本。我们的方法使用了晶格的新分解，这使我们能够编写表征步行的生成函数的功能方程，该步行功能按长度，端点和绕组角度计数。然后，我们根据jacobi theta函数求解这些功能方程。通过将此结果与反射原理结合使用，我们将步行限制在开放角度的圆锥体中，任何倍数的$ \fracπ{3} $，使我们能够为这些步行提取渐近和代数信息。我们的方法和结果类似地扩展到其他三个晶格，包括方格和三角形晶格。在广场晶格上，我们的大多数结果是由蒂莫西·布德（Timothy Budd）在2017年得出的，因此当前的工作可以将其视为Budd的结果扩展到我们考虑的其他三个晶格。 Budd推论这些结果的方法非常不同，因为它基于晶格中某些矩阵计数路径的明确特征值分解。
We address the problem of counting walks by winding angle on the Kreweras lattice, an oriented version of the triangular lattice. Our method uses a new decomposition of the lattice, which allows us to write functional equations characterising a generating function of walks counted by length, endpoint and winding angle. We then solve these functional equations in terms of Jacobi theta functions. By using this result in conjunction with the reflection principle, we count walks confined to a cone of opening angle any multiple of $\fracπ{3}$, allowing us to extract asymptotic and algebraic information for these walks. Our method and results extend analogously to three other lattices, including the square lattice and triangular lattice. On the square lattice, most of our results were derived by Timothy Budd in 2017, so the current work can be seen as an extension of Budd's results to the three other lattices that we consider. Budd's method of deducing these results was very different, as it was based on an explicit eigenvalue decomposition of certain matrices counting paths in the lattice.
