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I understand we have a fixed point in the couplings ("K") space (or in the scaling variable space). Then, there is a critical surface, which is attracted to it. This is a part of a system with some relevant variables, along with the irrelevant ones of the critical surface. The critical point in the system is attracted to the fixed point above. OK. Now, it is said in all the books that the fixed point "controls" the behavior in the critical point, specifically on the long distance behavior. What does it mean? Does it mean the critical exponents are the same in both points? Does it mean that something (what?) is identical for small k? Or what else?

thank you

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I would say that the fixed point "determines" the behaviour of the models on its critical surface in the sense that they have the same critical exponents. This is because critical exponents quantify behaviour at large scales, whereas the renormalization group transformation (by definition) only affects the model at microscopic scales (by integrating out the small-scale fluctuations).

One way to think of this is that large-scale behaviour is "tail" behaviour, as in, say, the tail of a sequence. For example, the convergence or divergence of an infinite series is not in determined by any finite number of terms in the series. More generally, the asymptotic properties of a sequence doesn't depend on finitely many terms of the sequence.

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