  In this study, we demonstrated experimentally that formation of chains and
islands of oxygen vacancies in hafnium sub-oxides (HfO$_x$, $x<2$) leads to
percolation charge transport in such dielectrics. Basing on the model of
\'{E}fros-Shklovskii percolation theory good quantitative agreement between the
experimental and theoretical data of current-voltage characteristics were
achieved. Based on the percolation theory suggested model shows that hafnium
sub-oxides consist of mixtures of metallic Hf nanoscale clusters of 1-2 nm
distributed onto non-stoichiometric HfO$_x$. It was shown that reported
approach might describe low resistance state current-voltage characteristics of
resistive memory elements based on HfO$_x$.
