Random eigenvalues of graphenes and the triangulation of plane Article Swipe
Artur Bille
,
Victor Matveevich Buchstaber
,
Simon Coste
,
Satoshi Kuriki
,
Evgeny Spodarev
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2306.01462
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2306.01462
We analyse the numbers of closed paths of length $k\in\mathbb{N}$ on two important regular lattices: the hexagonal lattice (also called $\textit{graphene}$ in chemistry) and its dual triangular lattice. These numbers form a moment sequence of specific random variables connected to the distance of a position of a planar random flight (in three steps) from the origin. Here, we refer to such a random variable as a $\textit{random eigenvalue}$ of the underlying lattice. Explicit formulas for the probability density and characteristic functions of these random eigenvalues are given for both the hexagonal and the triangular lattice. Furthermore, it is proven that both probability distributions can be approximated by a functional of the random variable uniformly distributed on increasing intervals $[0,b]$ as $b\to\infty$. This yields a straightforward method to simulate these random eigenvalues without generating graphene and triangular lattice graphs. To demonstrate this approximation, we first prove a key integral identity for a specific series containing the third powers of the modified Bessel functions $I_n$ of $n$th order, $n\in\mathbb{Z}$. Such series play a crucial role in various contexts, in particular, in analysis, combinatorics, and theoretical physics.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2306.01462
- https://arxiv.org/pdf/2306.01462
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
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Random eigenvalues of graphenes and the triangulation of planeWork title
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preprintOpenAlex work type
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enPrimary language
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2023Year of publication
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2023-06-02Full publication date if available
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Artur Bille, Victor Matveevich Buchstaber, Simon Coste, Satoshi Kuriki, Evgeny SpodarevList of authors in order
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https://arxiv.org/abs/2306.01462Publisher landing page
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https://arxiv.org/pdf/2306.01462Direct link to full text PDF
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2306.01462Direct OA link when available
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Mathematics, Lattice (music), Combinatorics, Eigenvalues and eigenvectors, Hexagonal lattice, Random variable, Bessel function, Discrete mathematics, Mathematical analysis, Physics, Quantum mechanics, Acoustics, Antiferromagnetism, StatisticsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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| abstract_inverted_index.straightforward | 124 |
| abstract_inverted_index.$k\in\mathbb{N}$ | 9 |
| abstract_inverted_index.$n\in\mathbb{Z}$. | 166 |
| abstract_inverted_index.$\textit{graphene}$ | 20 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 5 |
| citation_normalized_percentile |