Universal and non-universal signatures in the scaling functions of critical variables

TL;DR

This paper explores the interplay between universal and non-universal features in the scaling functions of critical variables, revealing that tail behaviors determine both universal exponents and non-universal amplitudes, supported by exact models.

Contribution

It demonstrates that tail behaviors of PDFs unify the understanding of universal exponents and non-universal amplitudes in critical phenomena.

Findings

Universal form of central limit theorem at criticality

Exact calculations for mean field Ising models

Confirmation in anomalous diffusion models

Abstract

The view that the probability density function (PDF) of a key statistical variable, anomalously scaled by size or time, could furnish a hallmark of universal behavior contrasts with the circumstance that such density sensibly depends on non-universal features. We solve this apparent contradiction by demonstrating that both non-universal amplitudes and universal exponents of leading critical singularities in large deviation functions are determined by the PDF tails, whose form is argued on extensivity. 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.