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THE LINKING PROBABILITY FOR 2-COMPONENT LINKS WHICH SPAN A LATTICE TUBE

M. ATAPOUR

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, Canada S7N 5E6, Canada

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C. E. SOTEROS

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, Canada S7N 5E6, Canada

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C. ERNST

Department of Mathematics and Computer Science, Western Kentucky University, Bowling Green, KY, USA 42101, USA

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, and

S. G. WHITTINGTON

Department of Chemistry, University of Toronto, Toronto, ON, Canada M5S 3H6, Canada

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https://doi.org/10.1142/S0218216510007760Cited by:14 (Source: Crossref)

Abstract

We consider two self-avoiding polygons (2SAPs) each of which spans a tubular sublattice of ℤ³. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. This pattern theorem is then used to prove that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover, we also use it to prove that the linking number Lk of an n edge 2SAP G_n satisfies lim_n→∞ℙ(|Lk(G_n)| ≥ f(n))=1 for any function . Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established that, due to the tube constraint, the linking number of an n edge 2SAP grows at most linearly in n.

Keywords:

AMSC: 57M25, 57M27

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