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Spin glass transitions of smectic-$A$ crosslinked elastomers

Elastomers are artificial polymeric materials created for industrial and commercial applications. Depending on their purpose, they are performing in different species and structure modifications. Our studies focus on the systems of elastomers randomly standing-distributed in a smectic $A$ (Sm$A$) liquid crystal. Basing on the suggestion following from the experiment, that at a phase transition from Sm$A$ to nematic phase caused by an increase of a crosslink concentration, such a system survives a percolation transition at low crossilink concentrations, we propose a modeling explaining this phenomena. We approve the three-dimensional Villain spin glass model and apply lattice Monte Carlo (MC) techniques on differential forms on a dual lattice, that is an alternative of a replica trick, developed for nematic elastomers in the 3D XY universality. In the results of that we have confirmed a concentration phase transition of percolation nature at a small crosslink concentration ($\sim$ 10 weight %).