  We prove that every 3-regular, n-vertex simple graph with sufficiently large
girth contains an independent set of size at least 0.4361n. (The best known
bound is 0.4352n.) In fact, computer simulation suggests that the bound our
method provides is about 0.438n.
  Our method uses invariant Gaussian processes on the d-regular tree that
satisfy the eigenvector equation at each vertex for a certain eigenvalue
\lambda. We show that such processes can be approximated by i.i.d. factors
provided that $|\lambda| \leq 2\sqrt{d-1}$. We then use these approximations
for $\lambda = -2\sqrt{d-1}$ to produce factor of i.i.d. independent sets on
regular trees.
