Invariant Gaussian processes and independent sets on regular graphs of large girth1. Issue 2 (16th May 2014) - Csóka, Endre; Gerencsér, Balázs; Harangi, Viktor; Virág, Bálint
Abstract 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhtsc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />λ</alternatives>. We show that such processes can be approximated by i.i.d. factors provided that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhttx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />|λ|≤2d−1</alternatives>. We then use these approximations for <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhtvg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />λ=−2d−1</alternatives> to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015