= Why microscopic differences fade near criticality
{title2=$\delta u_i(b)\simeq b^{y_i}\delta u_i(1)$}
Near a <renormalization-group fixed point>, expand a difference between two nearby models in <eigenvectors> of the <stability matrix>. An <irrelevant operator> direction has $y_i<0$, so its dimensionless coefficient shrinks under increasing length factor $b$. A <relevant operator> direction grows and must be tuned to study the same critical behavior. This is a mechanism for losing microscopic detail, not an assertion that every parameter disappears. Critical <temperature>, individual amplitudes and regular contributions to <free energy> can remain material dependent.
For the solvable example, write $A=\epsilon/g_0-1$. Its exact solution gives $g-\epsilon=-\epsilon A b^{-\epsilon}+O(b^{-2\epsilon})$. At the thermal stopping scale $b_*=|t_0|^{-1/y_t}$ this is of order $|t_0|^{\epsilon/y_t}$. If the stopped <correlation length> is smooth in $g$, the leading correction to its power law has that order. Thus two microscopic couplings can yield the same limiting <critical exponent>, yet approach it at different rates.
This mechanism makes <universality> useful: one tractable representative can predict <critical exponents> and normalized scaling forms for many systems in its <universality class>. A <liquid-gas critical point> and a scalar magnetic transition can share a long-distance description even though their microscopic constituents differ. The scope must be checked: spatial dimension, <order parameter> symmetry, interaction range, and extra soft variables can change the <renormalization-group fixed point>.
In his lecture, <Kenneth Geddes Wilson> connects limiting <renormalization-group fixed points> with common critical behavior and states the limits of his elementary truncation.<The Renormalization Group and Critical Phenomena by Kenneth Geddes Wilson>[Section II, pp. 110–112: truncation limits and common critical behavior]{ref}
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