For a relatively simple Markov chain Monte Carlo process, such as using Metropolis to find calculate thermal averages for an Ising model, how is it possible to determine whether quantities have converged?
If one knows the autocorrelation time, this seems relatively simple. You just run it for a sufficiently high multiple of this time and Bob's your uncle.
If you don't know the autocorrelation time, it would seem more complex. No matter how hard you try to ensure that your averages have converged, there's always the possibility that it is stuck in a local minima. This is especially troublesome if the autocorrelation time scales polynomially or even exponentially with the system size.
So how can you ensure that the quantities really have converged? What convergence tests are not fooled by local minima? I read here recently about the method of logarithmic binning. Could that do the job?