# The $\epsilon$ expansion and Universality in three dimensions

## TL;DR

This paper discusses how the classification of universality classes in critical phenomena, established via epsilon expansion near four or six dimensions, remains valid in three dimensions due to eigenvalue repulsion, despite failures of perturbative renormalization group.

## Contribution

It provides a theoretical argument that universality classification persists in lower dimensions through eigenvalue repulsion, beyond perturbative renormalization group methods.

## Key findings

- Universality classification remains valid in three dimensions.
- Eigenvalue repulsion explains the persistence of universality.
- Perturbative renormalization group fails in three dimensions, but universality persists.

## Abstract

It has been observed that the clasification into universality classes of critical behaviour, as established by perturbative renormalization group in the viscinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when purturbative renormalisation group fails in three dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when purturbative renormalisation group fails, because of the well known phenomenon of eigenvalue repulsion.