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≪N, its mean squared end to end distance growing as [Formula Presented]∝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≳(N/M[Formula Presented]. The number of links needed to collapse a polymer in three dimensions thus scales as [Formula Presented], with φ≳0.43.