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In this paper, we consider random iterations of polynomial maps $z^{2} +c_{n}$ where $c_{n}$ are complex-valued independent random variables followingthe uniform distribution on the closed disk with center $c$ and radius $r$. Theaim of this paper is twofold. First, we study the (dis)connectedness of randomJulia sets. Here, we reveal the relationships between the bifurcation radiusand connectedness of random Julia sets. Second, we investigate the bifurcationof our proposed random iterations and give quantitative estimates ofbifurcation parameters. In particular, we prove that for the central parameter$c = -1$, almost every random Julia set is totally disconnected with muchsmaller radial parameters $r$ than expected.

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$