Walks & Bloch band theory on periodic tessellations Davidson Noby Joseph

Reference

Phys. Rev. E
Walking on Archimedean lattices: Insights from Bloch band theory

Davidson Noby Joseph, and Igor Boettcher

Phys. Rev. E 112, 044118 – Published Oct 2025

Abstract arXiv Journal

Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As condensed matter physics applications, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices and analytically determine the asymptotics of the return probability. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.
@article{1fvj-91v6, title = {Walking on Archimedean lattices: Insights from Bloch band theory}, author = {Joseph, Davidson Noby and Boettcher, Igor}, journal = {Phys. Rev. E}, volume = {112}, issue = {4}, pages = {044118}, numpages = {29}, year = {2025}, month = oct, publisher = {American Physical Society}, doi = {10.1103/1fvj-91v6}, url = {https://link.aps.org/doi/10.1103/1fvj-91v6}, }

Imagine being lost in a maze, trying to find a way out by exploring different paths. You are particularly confused so you don’t hesitate to backtrack your steps. Your trajectory might look like the following

We identify the maze with a graph that contains vertices and edges; Each trajectory here is a walk which is a sequence of vertices through edges. Furthermore, you might’ve noticed that these trajectories actually end where they started – they are what’s called returning walks. As the number of possible steps increase, one can ask the following natural question: what fraction of paths return? The return probability $p_n$ is the fraction of paths that return to the start (called returning walk number) in $n$ steps. This lattice-dependent quantity can be estimated through variety of means, a brute force computation for $n=20$ yields

\[p_{20}=\frac{20,110,694,272}{4^20}=0.0182.\]

Would you believe it if this pure combinatorial number is directly related to $\pi$, the geometric constant? In the limit, we showed that this lattice (called the Maple-Leaf lattice), the return probability as $n\to \infty$ asymptotically scales as

\[p_{n} \sim \frac{1}{\pi} \frac{2}{\sqrt{3}} \frac{1}{n}.\]

Setting $n=20$, we find that

\[p_{2O} \sim \frac{1}{\pi} \frac{2}{\sqrt{3}} \frac{1}{20}=0.0183\approx 0.0182.\]

In our paper, we developed a framework to derive the asymptotic return probabilities for various periodic tessellations tested on a class of two-dimensional planar lattices called Archimedean lattices. In particular, we package these returning walk numbers into a generating function. Our formalism involves Bloch band theory using the notion of a Bloch generating function we introduce in our work. In addition, we used these generating functions to derive the density of states – an important physical observable – for the tight-binding model on these lattices and present a method to construct finite flakes of periodic lattices using graph-stitching.