 Why microscopic differences fade near criticality

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 $A=ϵ/g_0-1$. Its exact solution gives $g-ϵ=-ϵA{b}^{-ϵ}+O(b^{-2ϵ})$. At the thermal stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ this is of order $|t_0{|}^{ϵ/y_t}$. 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.^[ref]