As originally observed experimentally by Dehornoy, roots of Alexander polynomials of random knots display interesting patterns. In this work, joint with N. Dunfield, we prove several results on the distribution of such roots in the complex plane, and discuss further conjectures that originate from them.

Using the Burau representation, this corresponds to studying random walks on the group SL(2, C[t]) of 2-by-2 matrices with polynomial coefficients.

We compute a sharp lower bound on the probability that such roots lie on the unit circle, and prove a related central limit theorem. We also show there is a large root-free region near the origin.

We introduce the notion of a Lyapunov exponent for the Burau representation, in the spirit of Deroin-Dujardin, and a corresponding bifurcation measure, which we prove to be the limiting measure for the distribution of roots on a region of parameter space.