Random eigenvalues of graphenes and the triangulation of plane

TL;DR

The paper investigates the distribution of random eigenvalues for graphene and its planar triangulation by deriving explicit densities, generating functions, and characteristic functions that connect spectral density to planar random flights. It introduces a simple, simulatable approximation based on a uniform interval variable $X_b\sim U[0,b]$ and three irrational-frequency cosines, and proves convergence to the true distributions as $b\to\infty$, aided by a novel identity involving the cubes of modified Bessel functions $I_n$. The results unify combinatorial path counts on lattices with probabilistic representations, and provide constructive methods for sampling the random eigenvalues without explicit graph generation. The analysis leverages local weak convergence ideas, duality between hexagonal and triangulated lattices, and ergodic-theoretic arguments to illuminate the density of states in these planar carbon networks and to reveal deep connections between lattice geometry, special functions, and stochastic processes.

Abstract

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.

Figures (3)

• Figure 1: Illustration of the relationship between $\mathcal{H}$ and $\mathcal{T}$. The set $V(\mathcal{H})$ consists of both blue and red-colored vertices, whereas $V(\mathcal{T})$ contains only the red-colored ones. Edges of $\mathcal{H}$ and $\mathcal{T}$ are colored in black and green, respectively.
• Figure 2: Density functions $f_H$ (left) and $f_T$ (right) of the random eigenvalue $H$, $T$ of the hexagonal lattice $\mathcal{H}$ and triangular lattice $\mathcal{T}^*$, respectively.
• Figure 3: Density $f_T$ (red line) and normalized histograms of simulated $3+2Y_b$ (blue bars) for $b= 1$ (upper left), $b=10$ (upper right), $b=10^2$ (lower left), $b=10^5$ (lower right) with sample size $10^5$ and $\beta=\phi$.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Conjecture 1
• Theorem 1: Cotfas00, Theorem 3
• proof
• Theorem 2
• Proposition 1
• Proposition 2
• Proposition 3
• Remark 1