# Convergence of Non-Perturbative Approximations to the Renormalization Group

## TL;DR

This paper demonstrates that the derivative expansion in the non-perturbative renormalization group converges with a finite radius, provides guidelines for regulator selection, and shows rapid convergence of critical exponents in the 3D Ising model.

## Contribution

It proves the convergence of the derivative expansion and offers empirical rules for choosing optimal regulators in non-perturbative RG methods.

## Key findings

- Fast convergence of critical exponents in the 3D Ising model
- Finite radius of convergence for the derivative expansion
- Regulator choice does not affect convergence speed

## Abstract

We provide analytical arguments showing that the non-perturbative approximation scheme to Wilson's renormalisation group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of non-perturbative methods.