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

Published online by Cambridge University Press:  03 June 2015

Patrick W. Dondl ,
Michael Scheutzow  and
Sebastian Throm
Patrick W. Dondl
Affiliation:
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK, (patrick.dondl@durham.ac.uk)
Michael Scheutzow
Affiliation:
Fakultät II, Institut für Mathematik, Sekr. MA 7-5, Technische Universität Berlin, Strasse des 17 Juni 136, 10623 Berlin, Germany, (ms@math.tu-berlin.de)
Sebastian Throm
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany, (throm@iam.uni-bonn.de)
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 481 - 512
Copyright
Copyright © Royal Society of Edinburgh 2015 

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