Invariant Gaussian processes and independent sets on regular graphs of large girth

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.