Universality class

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In statistical mechanics, an universality class is a set of mathematical models which share a scale-invariant limit under renormalization group flow. While the models within a class may differ at finite scales, their behaviour become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents are the same for all models in the class. Well-studied examples include the universality classes of the Ising model or the percolation theory at their respective phase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes has a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2 for the Ising model, or for directed percolation, but 1 for undirected percolation), and above the upper critical dimension d ∗ {\displaystyle d^{*}} the critical exponents stabilize and can be calculated by an analog of mean-field theory (this dimension is d ∗ = 4 {\displaystyle d^{*}=4} for Ising or for directed percolation, and 6 for undirected percolation).