Conformations of randomly linked polymers

We consider polymers in which M randomly selected pairs of monomers are restricted to be in contact. Analytical arguments and numerical simulations show that an ideal (Gaussian) chain of N monomers remains expanded as long as M much less than N, its mean squared end to end distance growing as r(2) proportional to M/N. A possible collapse transition (to a region of order unity) is related to percolation in a one-dimensional model with long-ranged connections. A directed version of the model is also solved exactly. Based on these results, we conjecture that the typical size of a self-avoiding polymer is reduced by the links to R greater than or similar to(N/M)(nu). The number of links needed to collapse a polymer in three dimensions thus scales as N-phi, with phi greater than or similar to 0.43.