Ring Correlations in Two-Dimensional (2D) Random Networks

Amorphous materials can be characterized by their ring structure. Recently, two experimental groups imaged bilayers of vitreous silica at atomic resolution which provides a direct access to the ring structure of a 2D glass \footnote{Lichtenstein L et al, \textit{Angew. Chem. Int. Ed.} 51 404 (2012) and Huang P Y et al, \textit{Nano Lett.} 12 1081 (2012)}. It has been shown that experimental samples have various ring statistics, obey Aboav-Weaire law and have a distinct area law \footnote{Kumar A et al, \textit{Journal of Physics: Condensed Matter} 26 39 (2014): 395401.}. In this work, we study correlations between rings as a function of their size and topological separation. We show that correlation is medium-range and vanishes when the separation is about three rings apart. We also present a generalization of the Aboav-Weaire law.