To decorrelate a configuration, need to make changes on a scale > spatial correlation length. For local MC algorithms (such as Metropolis) the effect of a change is only seen by nearest neighbors. Since the update is a stochastic process, the changes should propagate through the system like a random walk, so the number of iterations required to propagate a distance goes like . At the critical point, , so we expect . This is a major problem for MC simulations, since we need large L.
More generally, where z is called the dynamic critical exponent. For a local algorithm and usually . One of the active areas of research in MC simulation is trying to find new algorithms with z<2. These are usually non-local, multi-scale algorithms, c.f. multigrid methods for solving PDEs, where the same type of problem occurs.