Abstract
The most exciting moment we encounter while studying mathematics is when we observe that a seemingly isolated subject turns out to be connected with other fields in an unexpected way. In the present and next chapters, we shall give two such examples in connection with standard realizations. The first is asymptotic analysis of random walks on topological crystals which motivated the author to introduce the concept of standard realization [58, 60], and is the theme of this chapter. The second, to be explained in the next chapter, is a discrete (combinatorial) analogue of classical algebraic geometry, a field of more recent vintage, in which the standard realizations in a processed form show up as an analogue of the Abel–Jacobi map.
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Notes
- 1.
See Woess [112].
- 2.
It is not a difficult task to establish the central limit theorem if we do not care about the explicit and geometric shape of the Gaussian function involved in the formula.
- 3.
Strictly speaking, this asymptotic expansion holds in the non-bipartite case since, in the bipartite case, we have p(n, x, y) = 0 if x, y have different colors and n is even. Thus a slight modification is required in the bipartite case. See Sect. 9.5.
- 4.
A unitary character of L is a one-dimensional unitary representation of L, i.e., a homomorphism of L into \(U(1) =\{ z \in \mathbb{C}\vert \ \vert z\vert = 1\}\). The set of all unitary characters has a natural group structure.
- 5.
A finite-dimensional vector space with an inner product.
- 6.
\({P}_{\mathbf{x}} - I\) is a discrete analogue of the twisted Laplacian [91].
- 7.
See Theorem 7.7.5 (p. 220) in Hörmander [49].
- 8.
The converse is also true. See Notes (II) in this chapter.
- 9.
A topological group is a topological space with a group structure such that group operations (product and inverse) are continuous. As a topological space, Aut(X) is totally disconnected.
- 10.
When X is bipartite, we take bipartite graphs X 0 and X 0′ as base graphs.
- 11.
Roughly speaking, maximal symmetry means that no structural deformation of its periodic arrangement of atoms in a crystal will make the structure more symmetrical than it is.
- 12.
In general, f(n) = O(n − k) means that \(\vert f(n)\vert \leq A{n}^{-k}\) for some constant A.
- 13.
This is a disguised form of Bloch–Floquet theory applied to the operator P. As for Bloch–Floquet theory for differential operators, see [64].
- 14.
For a hermitian operator \(T : H\rightarrow H\) of a Hilbert space H, the spectrum of T, symbolically Spect(T), is the set of complex numbers λ such that T − λI is not invertible as a bounded operator. It is known that Spect(T) is a bounded closed subset of \(\mathbb{R}\).
- 15.
As usual, in bipartite case, n runs over even numbers when x and y have the same color, or odd numbers when x,y have different colors.
- 16.
A higher dimensional analogue of convex polygons in plane and polyhedra in space.
- 17.
The “physical” continuum limit of the net corresponding to a real crystal is an elastic body (cf. [85]).
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Sunada, T. (2013). Random Walks on Topological Crystals. In: Topological Crystallography. Surveys and Tutorials in the Applied Mathematical Sciences, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54177-6_9
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