Local algorithms for independent sets are half-optimal

We show that the largest density of factor of i.i.d. independent sets on the d-regular tree is asymptotically at most (log d)/d as d tends to infinity. 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 on these graphs is asymptotically 2(log d)/d. We also prove analogous results for Poisson-Galton-Watson trees, which yield bounds for local algorithms on sparse Erdos-Renyi graphs.