Local nearest-neighbor correlation as a critical indicator: Universality-class fingerprints and nonequilibrium dynamics

Critical phenomena are traditionally characterized by global order parameters and their fluctuations, yet local correlations carry information about universality classes that global averages suppress. We introduce a local-order purity P defined via the absolute nearest-neighbor correlation ⟨|σiσj|⟩, distinct from the standard correlation u=⟨σiσj⟩. The quantity P captures non-Gaussian fluctuation effects through the rigorous bound Δ≥|u|. For the 2D and 3D Ising universality classes, P takes universal critical values Pc(2D)≈0.517 and Pc(3D)≈0.36–0.38, serving as class-specific fingerprints. In the 2D XY model, P(T) shows a slope enhancement near the Kosterlitz-Thouless temperature where the Binder cumulant is featureless, providing a low-cost complementary diagnostic. Under Kawasaki dynamics with conserved magnetization, P(t) exhibits depth-dependent four-stage coarsening. The locking time follows τlock∝|Tc−Tq|−α with α≈1.73 across four quench depths. The cumulative information dissipation εin(t) decomposes into linear drift, exponential relaxation, and long-range correlated noise (R2>0.999). The quantity εin correlates strongly with domain-wall activity (r>0.79) but only weakly with thermodynamic entropy production, constituting an operationally defined measure of order-parameter-space roughness. These results establish P as a computationally efficient, locally defined critical indicator applicable to both equilibrium and nonequilibrium statistical mechanics.