critical exponents and scaling in percolation

Near the critical point percolation has no characteristic length scale, and physical quantities behave as power laws of the distance to criticality. The exponents in those power laws — collectively the critical exponents — are the fingerprints of the transition, and the remarkable empirical fact is that they are UNIVERSAL: they depend only on the dimension d, not on the fine details of the lattice (square vs triangular, bond vs site). Scaling theory is the framework that organizes these exponents and relates them by a few master relations.

The standard exponents are defined by power laws as p approaches p_c. The order parameter vanishes as theta(p) ~ (p - p_c)^beta for p > p_c; the mean cluster size (expected size of the finite cluster) diverges as chi(p) ~ |p - p_c|^(-gamma); the correlation length diverges as xi(p) ~ |p - p_c|^(-nu); at criticality the cluster-size distribution has a power-law tail P_{p_c}(|C| >= k) ~ k^(-1/delta'), and two-point connectivity at p_c decays as a power of distance, P_{p_c}(0 connects to x) ~ |x|^(-(d-2+eta)). Scaling theory posits that near p_c there is a single diverging length xi controlling everything, so all observables are homogeneous functions of (p - p_c) and the system size measured in units of xi. This forces scaling relations among the exponents, e.g. gamma = nu(2 - eta) (Fisher) and the hyperscaling relation d nu = 2 beta + gamma, which involves the dimension explicitly. In two dimensions the exponents are known exactly (and rational): beta = 5/36, gamma = 43/18, nu = 4/3, eta = 5/24, delta = 91/5, established via the SLE/conformal-invariance program.

Two honesty points. First, hyperscaling (the relation containing d) holds only BELOW the upper critical dimension d_c = 6; for d > 6 the exponents stick at their mean-field (Bethe-lattice) values beta = 1, gamma = 1, nu = 1/2, delta = 2, and hyperscaling fails because the critical cluster is effectively a tree that does not 'see' the ambient dimension. Hara and Slade proved the mean-field values rigorously for d >= 11 (later improved toward 6) via the lace expansion. Second, the EXISTENCE of these exponents as honest limits is itself often a hard theorem; physicists assume the power laws, but rigorously establishing even that the limits exist (let alone their values) is open in dimensions 3, 4, 5 as of the early 2020s. Universality, while overwhelmingly supported, is largely conjectural outside two dimensions and the mean-field regime.