# Universality of scaling of correlations across probability distributions

## TL;DR

This paper demonstrates that the scaling behavior of correlations in translation, rotational, and scale invariant lattice systems with arbitrary probability distributions aligns with that of critical equilibrium models, extending universality concepts beyond Boltzmannian distributions.

## Contribution

It establishes a universal link between correlation scaling exponents and critical statistical models for systems with arbitrary probability distributions.

## Key findings

- Correlation exponents match those of critical Boltzmannian models.
- Universality extends to non-Boltzmann probability distributions.
- Scaling behavior is consistent across diverse probability distributions.

## Abstract

Scale invariance and the resulting power law behaviours are seen in diverse systems. In this work we consider translation, rotational and scale invariant systems defined on a lattice, such that the variables defining the state at every lattice site take on the same range of finite values, with these values collectively picked up from probability distribution that can be arbitrary. We show that the exponent that describes the scaling of the two point correlation function in these systems will match the scaling exponent of a equilibrium statistical mechanical model described by a Boltzmannian distribution at criticality. This work therefore extends the concept of universality in statistical mechanics to probability distributions that do not have a Boltzmannian form.