  In this paper, we study random embeddings of polymer networks distributed
according to any potential energy which can be expressed in terms of distances
between pairs of monomers. This includes freely jointed chains, steric effects,
Lennard-Jones potentials, bending energies, and other physically realistic
models.
  A configuration of $n$ monomers in $\mathbb{R}^d$ can be written as a
collection of $d$ coordinate vectors, each in $\mathbb{R}^n$. Our first main
result is that entries from different coordinate vectors are uncorrelated, even
when they are different coordinates of the same monomer. We predict that this
property holds in realistic simulations and in actual polymer configurations
(in the absence of an external field).
  Our second main contribution is a theorem explaining when and how a
probability distribution on embeddings of a complicated graph may be pushed
forward to a distribution on embeddings of a simpler graph to aid in
computations. This construction is based on the idea of chain maps in homology
theory. We use it to give a new formula for edge covariances in phantom network
theory and to compute some expectations for a freely-jointed network.
