We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model. We show that the sequence of manifolds converges to a smooth Riemannian manifold, while the Levi-Civita connections converge to a non-metric connection on the limit manifold. Thus, we obtain rigorously the emergence of a non-metricity tensor, which was postulated in the literature to represent continuous distribution of point defects.
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Acknowledgements. We are very grateful to Marcelo Epstein for suggesting the question of homogenization of defects, and to Pavel Giterman for fruitful discussions. We are also grateful to the anonymous referees, who pointed out some errors and helped us to improve the readability of the paper. The first author is partially supported by the Israel Science Foundation and by the Israel– US Binational Foundation. The third author is partially supported by an ETH fellowship.
© 2018, Hebrew University of Jerusalem.