Local algorithms for independent sets are half-optimal

We show that the largest density of factor of i.i.d. independent sets in the d-regular tree is asymptotically at most (logd)/d as d... This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random d-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically 2(logd)/d. We prove analogous results for Poisson.Galton.Watson trees, which yield bounds for local algorithms on sparse Erd.s.Rényi graphs.