Universal and Nonuniversal Signatures in the Scaling Functions of Critical Variables

Area of Science:

• Statistical Physics
• Complex Systems

Background:

• At criticality, the magnetization (M) of d-dimensional Ising systems with N spins follows a probability density function f(m), where m=M/N^{y_{H}/d}.
• The universality of f(m) is often questioned due to its dependence on nonuniversal features.

Purpose of the Study:

• To investigate the universal behavior of the probability density function f(m) at criticality.
• To demonstrate that universal exponents and nonuniversal amplitudes of large deviation functions are determined by extensivity.

Main Methods:

• Analysis of the large deviation functions for the probability density function f(m).
• Exact calculations for mean-field Ising models and anomalous diffusion models.

Main Results:

• Demonstrated that f(m) exhibits power-law singularities with universal exponents and nonuniversal amplitudes.
• Showed that f(m)∼_{|m|≫1}exp(-c|m|^{δ+1}), with δ=y_{H}/(d-y_{H}).
• Confirmed these findings through exact calculations.

Conclusions:

• The study implies a universal form of the central limit theorem at criticality.
• The findings hold for both equilibrium Ising models and anomalous diffusion processes.