# Asymptotically Exact Solution of the Fredrickson-Andersen Model

## TL;DR

This paper provides an asymptotically exact analytical solution for the dynamical behavior of the Fredrickson-Andersen model on the Bethe lattice, revealing critical exponents and proposing an approximate dynamics for broader regimes.

## Contribution

It derives an exact expression for the memory kernel of the FA model, enabling explicit calculation of critical exponents and offering an approximate dynamics description.

## Key findings

- Exact solution for the memory kernel of the FA model.
- Explicit critical exponents satisfying a scaling relation.
- Approximate dynamics matching numerical data away from criticality.

## Abstract

The Fredrickson-Andersen (FA) model---a kinetically constrained lattice model---displays an ergodic to non-ergodic transition with a slow two-step relaxation of dynamical correlation functions close to the transition point. We derive an asymptotically exact solution for the dynamical occupation correlation function of the FA model on the Bethe lattice by identifying an exact expression for its memory kernel. The exact solution fulfills a scaling relation between critical exponents and allows to calculate the exponents explicitly. In addition, we propose an approximate dynamics that describes numerical data away from the critical point over many decades in time.