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We prove that every 3‐regular,  n ‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361 n . (The best known bound is 0.4352 n .) In fact, computer simulation suggests that the bound our method provides is about 0.438 n .  Our method uses invariant Gaussian processes on the  d ‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue   λ  . We show that such processes can be approximated by i.i.d. factors provided that    |  λ  |  ≤  2    d  −  1. We then use these approximations for    λ  =  −  2    d  −  1to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015

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