Limiting measure of roots for random positive 3-braids

Establish the existence of a compactly supported probability measure ν∞ on the complex plane C such that, for the random walk w_n = g_1 g_2 ··· g_n in the 3-strand braid group Br_3 generated by independent, uniformly distributed choices g_i ∈ {σ_1, σ_2}, the empirical root measures ν_{w_n} of the Alexander polynomials of the closures converge weakly almost surely to ν∞, independently of the sample path. Determine further that: (i) the support of ν∞ lies in the union of the unit circle, a continuous curve connecting the cubic roots of unity ζ_3 and ζ̄_3, and its image under inversion z ↦ 1/z, with the curve crossing the real axis near −0.78; (ii) exactly 2/3 of the mass of ν∞ lies on the unit-circle arc A_R = {e^{iθ}: |θ|<2π/3}, where ν∞ is a multiple of Lebesgue measure, and ν∞ is absolutely continuous on the complementary arc A_L = {e^{iθ}: |θ−π|<π/3} with total mass (7−3√5)/12; and (iii) along the interior curves from ζ_3 to ζ̄_3, ν∞ is absolutely continuous with respect to arc length.

Background

The paper studies the roots of Alexander polynomials of links obtained as closures of positive 3-strand braids, focusing on statistical behavior for random braids under the uniform measure on {σ_1, σ_2}. Empirical evidence shows roots concentrate on the unit circle and on a single interior curve from ζ_3 to ζ̄_3, with approximately 69% on the unit circle and an apparent equidistribution along certain arcs.

Motivated by bifurcation currents and Lyapunov exponents of the Burau representation, the authors propose a precise limiting measure ν∞ for the root distributions, including detailed support and mass distribution properties. They prove several partial results—such as lower bounds on the proportion of roots on A_R and equidistribution on subsets—while leaving the full convergence and characterization open.