Sweeping words and the length of a generic vector subspace of Mn(F)Mn(F)

The main result of this short note is a generic version of Paz' conjecture on the length of generating sets in matrix algebras. Consider a generic g  -tuple A_=(A1,…,Ag) of n×nn×n matrices over an infinite field. We show that whenever g2d≥n2g2d≥n2, the set of all words of degree 2d   in A_ spans the full n×nn×n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of special kinds of directed multigraphs with few edge-disjoint walks.