Violent fluctuations of the fracton density of states on the percolation cluster and its backbone.

Abstract

We report the fracton density of states (DOS) and its fluctuation properties on the infinite two-dimensional critical percolation cluster and its backbone. The fracton DOS fluctuations, as expressed by the number variance 〈[\ensuremath{\delta}N(E)${]}^{2}$〉 in an energy width E, follow the quadratic law 〈[\ensuremath{\delta}N(E)${]}^{2}$〉\ensuremath{\propto}〈N(E)${\mathrm{〉}}^{2}$, instead of the usual linear Poissonic behavior normally expected for localized states. We also find that the average DOS for the percolation backbone obeys the one-parameter fracton scaling theory with a spectral dimension ${\mathit{d}}_{\mathit{s}}^{\mathrm{BB}}$=1.23\ifmmode\pm\else\textpm\fi{}0.02. This kind of violent DOS fluctuation cannot be understood in the context of random matrix theories and is discussed in connection to intermittency and multifractal localization.