Universality class

Systems belong to one universality class when their long-distance renormalization-group flows approach the same fixed point.

Under the scaling hypothesis for critical phenomena, $F_s=A_f|a_{t}t{|}^{D/y_t}{\Phi }_{\pm }(0)$ at zero field. The common free-energy and thermal metric factors cancel from $A_{+}/A_{-}$, leaving a property of the universality class. This concerns the singular part at equal $|t|$, not the ratio of full free energies including arbitrary analytic identity terms. Matching constants in ${\Phi }_{\pm }$ belong to the same scaling solution and cannot be assigned independently.

When the singular free-energy density has $F_s=A_f|a_{t}t{|}^{2-\alpha }{\Phi }_{\pm }(0)$, the same thermal and free-energy metric factors multiply the singular heat capacity above and below the critical point. Their ratio cancels these factors and depends only on the normalized scaling functions of the universality class. This statement concerns singular amplitudes, not regular heat-capacity backgrounds, and needs separate treatment at logarithmic resonances or zero amplitudes. Branch matching constants belong to the fixed-point scaling functions and cannot be chosen as independent nonuniversal parameters.

Ordinary short-range independent bond percolation and site percolation on regular lattices of the same spatial dimension are expected to share their percolation critical exponents and large-scale limiting behavior. Their percolation critical probabilities can differ. Long-range or correlated models can lie in a different universality class. A theorem for one lattice alone does not establish this transfer.