Asymptotics of multicomponent linked polygons

We investigate the asymptotic behaviour of multi-component links where the edges can be distributed among the components in all possible ways. Specifically we consider a link of k polygons on the simple cubic lattice. We prove two results about the exponential behaviour and use a Monte Carlo method to investigate how the value of the critical exponent depends on link type. One ring grows at the expense of the others while the remaining components act as one or more roots on the growing component, each root contributing 1 to the value of the critical exponent. Which component grows depends on which maximizes the entropy of the system