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We study Markov processes in which $$\pm 1$$ -valued random variables $$\sigma _x(t), x\in \mathbb {Z}^d$$ , update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $$\rho ^+$$ (resp., $$\rho ^-$$ ), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $$\rho ^+ >0$$ and $$\rho ^- =0$$ , all sites are fixed plus, while for $$\rho ^+ >0$$ and $$\rho ^-$$ very small (compared to $$\rho ^+$$ ), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices. [ABSTRACT FROM AUTHOR]

The deviation from stoichiometry and the understanding of its consequences are key factors for the application of kesterites as solar cell absorbers. Therefore, this study investigates the local atomic structure of off-stoichiometric Cu2ZnSnS4 (CZTS), Cu2ZnSnSe4 (CZTSe) and Cu2ZnGeSe4 (CZGSe) by means of Extended X-ray Absorption Fine Structure Spectroscopy. Temperature dependent measurements yield the bond stretching force constants of all cation-anion bonds in stoichiometric CZTS and CZTSe and nearly stoichiometric CZGSe. Low temperature measurements allow high precision analysis of the influence of off-stoichiometry on the element specific average bond lengths and their variances. The overall comparison between the materials is in excellent agreement with measures like ionic/atomic radii and bond ionicities. Furthermore, the small uncertainties allow the identification of systematic trends in the Cu-Se and Zn-Se bond lengths of CZTSe and CZGSe. These trends are discussed in context of the types and concentrations of certain point defects, which gives insight into the possible local configurations and their influence on the average structural parameters. The findings complement the understanding of the effect of off-stoichiometry on the local structure of kesterites, which affects their electronic properties and thus their application for solar cells., (© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).)