Abstract
Using operators and elementary Fourier methods, we analyze walks in one-dimensional bounded and unbounded lattices.
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Notes
- 1.
We assume an elementary knowledge of linear algebra, including the definition of a vector space. For review, any undergraduate text in linear algebra would suffice, e.g., [39, 40, 48].
- 2.
Specifically, \(T = R + L\) will be called the classical or two-way transition operator.
- 3.
See Appendix A for a refresher on complex numbers and the complex exponential function.
- 4.
For a recent explanation and use of this technique, see [49].
- 5.
OEIS sequence A000984.
- 6.
OEIS sequence A000108 in the OEIS. See also Exercise 1.7.
- 7.
Indeed, \(v_0\) is the m-periodization of \(\delta \) as defined in Eq. (2.11).
- 8.
See OEIS sequences A078008, A199573, A054877, A047849, A094659.
- 9.
According to Pólya’s theorem, this is also true of lattice walks in dimension 2. That is, if the “drunkard” is walking wandering around the streets of an infinite city grid, the drunkard will almost surely return to the same bar infinitely often, so to speak. However, in higher dimensions, the drunkard becomes hopelessly lost; i.e., the walk will remain outside any given ball, with probability 1 (see also [21]). This result is not just an idle curiosity; there are also interesting connections between random walks and electrical engineering [19, 62].
- 10.
Using a fast matrix multiplication algorithm, the time complexity for finding the determinant of an \(n \times n\) matrix is roughly \(O(n^{2.373})\)Â [2].
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Ault, S., Kicey, C. (2019). One-Dimensional Lattice Walks. In: Counting Lattice Paths Using Fourier Methods. Applied and Numerical Harmonic Analysis(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26696-7_2
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