I think you should look at this paper "The average crossing number of equilateral random polygons" by Y Diao, A Dobay, R B Kusner, K Millett and A Stasiak. http://iopscience.iop.org/0305-4470/36/46/002

It can't seem to get the full text right now, so I'm not sure exactly what their random knot model is. It certainly isn't quite what you wanted, as they require their polygons to be equilateral. In their model the crossing number grows as $n \; \ln n$.

If this doesn't answer your question, consider looking through the rest of Ken Millett's work. He is one of the main people working on random knots.

I think you should look at this paper: Diao, Y.; Dobay, A.; Kusner, R. B.; Millett, K.; Stasiak, A., The average crossing number of equilateral random polygons, J. Phys. A, Math. Gen. 36, No. 46, 11561–11574 (2003). Zbl 1052.57006.

It can't seem to get the full text right now, so I'm not sure exactly what their random knot model is. It certainly isn't quite what you wanted, as they require their polygons to be equilateral. In their model the crossing number grows as $n \; \ln n$.

If this doesn't answer your question, consider looking through the rest of Ken Millett's work. He is one of the main people working on random knots.