Universal and Nonuniversal Signatures in the Scaling Functions of Critical Variables

Abstract

At criticality, the magnetization M of a d-dimensional Ising system with N spins is distributed such that the probability density function of m=M/NyH/d, with yH universal exponent, converges to a limit f(m). The expectation that f should furnish a hallmark of universal behavior contrasts with its sensible dependence on nonuniversal features. We show that both nonuniversal amplitudes and universal exponents of leading power law singularities in all large deviation functions are determined by the fact that, due to extensivity, f(m)∼|m|≫1exp(−c|m|δ+1), with δ=yH/(d−yH) and c a nonuniversal coefficient. This unexplored scenario implies a universal form of central limit theorem at criticality and is confirmed by exact calculations for mean field Ising models in equilibrium and for anomalous diffusion models, with M replaced by displacement and N by time. Published by the American Physical Society 2025