We study the expansion of the surface thickness in the 2-dimensional l
attice sine Gordon model in powers of the fugacity z. Using the expans
ion to order z2, we derive lines of constant physics in the rough phas
e. We describe and test a VMR cluster algorithm for the Monte Carlo si
mulation of the model. The algorithm shows nearly no critical slowing
down. We apply the algorithm in a comparison of our perturbative resul
ts with Monte Carlo data.