Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

Dondl, Patrick W.; Scheutzow, Michael; Throm, Sebastian

Abstract

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

Citation

Dondl, P. W., Scheutzow, M., & Throm, S. (2015). Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 145(3), 481-512. https://doi.org/10.1017/s0308210512001291

• Journal Article Type: Article
• Acceptance Date: Jan 24, 2014
• Online Publication Date: Jun 3, 2015
• Publication Date: Jun 3, 2015
• Deposit Date: Oct 24, 2014
• Publicly Available Date: Jun 15, 2015
• Journal: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
• Print ISSN: 0308-2105
• Electronic ISSN: 1473-7124
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 145
• Issue: 3
• Pages: 481-512
• DOI: https://doi.org/10.1017/s0308210512001291
• Public URL: https://durham-repository.worktribe.com/output/1420998
• Related Public URLs: http://arxiv.org/abs/1201.4836