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 → ∞. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random dregular 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 Erdős-Rényi graphs.