# Why are the autocorrelations larger for the energy at the critical temperature?

Considering a simulation with the Swendsen-Wang algorithm for the 3-D cubic lattice I wanted to have a look at the auto-correlations, and expecting it to be quite small considering Swendsen-Wang is a cluster algorithm.

The results are as expected, except that at the critical temperature the autocorrelation for the energy is much larger than for other temperatures.

Is this expected, and if so, what are the reasons behind this?

Using the renormalization group (see, for example, chapter 3 of Cardy's book), it is possible to prove that the correlation length $$\xi(T)$$ of your spin system diverges at the critical temperature $$T_c$$; as a result, the spin-spin correlator $$G(r)$$ as a function of the spin-spin distance $$r$$ goes to zero with a power law $$G(r) \sim r^{-(D - 2 + \eta)}$$ instead of going to zero with the usual exponential law $$G(r) \sim r^{-(D - 2 + \eta)}e^{-r/\xi},$$ i.e, the spin-spin correlation goes to zero much slower at $$T = T_c$$. Here $$D$$ are the dimensions of the model and $$\eta$$ is a critical exponent depending on the universality class.
Therefore, out-of-equilibrium fluctuations turn out to be qualitatively very larger than usual in a Monte Carlo update algorithm, and it takes longer to perform two indipendent measures of (for example) the energy of the system. Quantitatively, the autocorrelation time and correlation length depend on each other in the following way: $$\tau_a \sim \xi^z,$$ where $$z = 2$$ for local updating algorithms.