  A theorem of Hoffman gives an upper bound on the independence ratio of
regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the
adjacency matrix. To complement this result we use random eigenvectors to gain
lower bounds in the vertex-transitive case. For example, we prove that the
independence ratio of a $3$-regular transitive graph is at least
\[q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda
_{\min}}{4}\biggr).\] The same bound holds for infinite transitive graphs: we
construct factor of i.i.d. independent sets for which the probability that any
given vertex is in the set is at least $q-o(1)$. We also show that the set of
the distributions of factor of i.i.d. processes is not closed w.r.t. the weak
topology provided that the spectrum of the graph is uncountable.
