  We describe a model of random links based on random 4-valent maps, which can
be sampled due to the work of Schaeffer. We will look at the relationship
between the combinatorial information in the diagram and the hyperbolic volume.
Specifically, we show that for random alternating diagrams, the expected
hyperbolic volume is asymptotically linear in the number of crossings. For
nonalternating diagrams, we compute the probability of finding a given,
arbitrary tangle around a given crossing, and show that a random link diagram
will be highly composite. Additionally, we present some results of computer
experiments obtained from implementing the model in the program SnapPy.
