Matrix random products with singular harmonic measure

Any Zariski dense countable subgroup of SL(d, mathbb R){SL(d, mathbb {R})} is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in mathbb R^d{mathbb {R}^d} in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and arbitrarily long Lyapunov vector.