We consider a model of 1D multimodal circle maps with strong expansion on most of phase space, including, e.g., the one-parameter family fa(x)=Lsinx+a for a∈[0,1) with fixed L>>1. Even when L is quite large, the problem of deciding the asymptotic regime (stochastic versus regular) of fa for a given a involves infinite-precision knowledge of infinite trajectories: outside special cases, this problem is typically impossible to resolve from any checkable finite-time conditions on the dynamics of fa. We contend that the corresponding problem for (possibly quite small) IID random perturbations of the fa is far more tractable. In our model, we perturb fa at each timestep by an IID uniformly distributed random variable in the interval [−ϵ,ϵ] for a fixed (yet arbitrarily small) ϵ>0. We obtain a checkable condition, involving finite trajectories of fa of length ~ log(ϵ−1), for this random composition to admit (1) a unique, absolutely continuous stationary ergodic measure and (2) a Lyapunov exponent of size approximately logL. Joint with Yun Yang.