Mean Field Behavior of Cluster Dynamics

TL;DR

This paper analyzes the dynamic behavior of cluster algorithms in the mean field limit, providing rigorous results for their dynamic exponents and demonstrating their scaling properties through extensive simulations.

Contribution

It offers the first rigorous analytical determination of dynamic exponents for Swendsen-Wang and Wolff algorithms in the mean field limit and introduces an efficient Monte Carlo implementation.

Findings

Swendsen-Wang algorithm has dynamic exponent z=1 below T_c

Wolff algorithm has dynamic exponent z=0 below T_c

Finite-size scaling functions show impressive data collapse

Abstract

The dynamic behavior of cluster algorithms is analyzed in the classical mean field limit. Rigorous analytical results below $T_c$ establish that the dynamic exponent has the value $z_{sw}=1$ for the Swendsen-Wang algorithm and $z_{uw}=0$ for the Wolff algorithm. An efficient Monte Carlo implementation is introduced, adapted for using these algorithms for fully connected graphs. Extensive simulations both above and below $T_c$ demonstrate scaling and evaluate the finite-size scaling function by means of a rather impressive collapse of the data.