Random eigenvalues of graphenes and the triangulation of plane

Abstract We analyze the numbers of closed paths of length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> on two important regular lattices: the hexagonal lattice (also called 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 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> . 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . Such series play a crucial role in various contexts, in particular, in analysis, combinatorics, and theoretical physics.