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Existence of a phase transition for entanglement percolation

Published online by Cambridge University Press:  16 October 2000

ALEXANDER E. HOLROYD
ALEXANDER E. HOLROYD
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095, U.S.A.; e-mail: [email protected]
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Abstract

We consider the bond percolation model on the three-dimensional cubic lattice, in which individual edges are retained independently with probability p. We shall describe a subgraph of the lattice as ‘entangled’ if, roughly speaking, it cannot be ‘pulled apart’ in three dimensions. We shall discuss possible ways of turning this into a rigorous definition of entanglement. For a broad class of such definitions, we shall prove that for p sufficiently close to zero, the graph of retained edges has no infinite entangled subgraph almost surely, thereby establishing that there is a phase transition for entanglement at some value of p strictly between zero and unity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 2 , September 2000 , pp. 231 - 252
Copyright
2000 Cambridge Philosophical Society

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