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The thing is that I don't understand how it is possible that in second order phase transitions the correlation legth

becomes infinite. I don't understand the scale invariance of thermal fluctuations in these critical phenomena. How is this fact consistent with special relativity? After all a local perturbation should never be able to propagate faster than the speed of light, right?

There's something that I am not getting right. But the more I think and read about this the less I understand it.

Thanks in advance for your help.