Random eigenvalues of nanotubes
Details
The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in graphene and carbon $(p,q)$-nanotubes. The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths $k\in\mathbb{N}_0$ on their respective graph representations. Considering that a carbon $(p,q)$-nanotube can be thought of as a graphene sheet rolled up in a matter determined by the chiral vector $(p,q)$, our findings are based on the study of random eigenvalues of both the hexagonal and triangular lattices presented in [bille2023random]. This study reveals that for any given chiral vector $(p,q)$, the sequence of counts of closed paths forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including probability density function (PDF) and moment generating function (MGF), are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the circumference of a $(p,q)$-nanotube approaches infinity, i.e., $p+q\rightarrow \infty$, the $(p,q)$-nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length $k$, indicating weak convergence of the underlying distributions.
Citation
BibTeX
@article {MR4887136,
AUTHOR = {Bille, Artur and Buchstaber, Victor and Ievlev, Pavel and Novikov, Svyatoslav and Spodarev, Evgeny},
TITLE = {Random eigenvalues of nanotubes},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics. A. Mathematical and Theoretical},
VOLUME = {58},
YEAR = {2025},
NUMBER = {10},
PAGES = {Paper No. 105202, 34},
ISSN = {1751-8113},
MRCLASS = {05C50 (33C10 81Q35)},
MRNUMBER = {4887136}
}