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The Wikipedia entry for scale invariance states

In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena.

I assume the implicit rationale behind "near a phase transition or critical point, fluctuations occur at all length scales" is that the at the critical point, a system's correlation length diverges.

In general, the correlation length will not be the only scale within a theory, though. There could be masses, charges, etc. that introduce a scale as well.

Why is it still a universally valid statement that physics becomes scale independent at phase transitions?

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In the types of system in which second-order phase transitions are studied, forces are generally short range. This means that the other scales you mention will be finite in size. However, at the critical point the correlation length diverges, which means it becomes effectively infinite. If you look at the system on larger and larger scales, any finite scale will eventually become a microscopic detail, and the system's behaviour becomes dominated by the scale-free critical point behaviour.

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