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

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 Gnsatisfies limn→∞ℙ(|Lk(Gn)| ≥ f(n))=1 for any function [Formula: see text]. 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.