  We study the expansion of the surface thickness in the 2-dimensional lattice
Sine Gordon model in powers of the fugacity z. Using the expansion to order
z**2, we derive lines of constant physics in the rough phase. We describe and
test a VMR cluster algorithm for the Monte Carlo simulation of the model. The
algorithm shows nearly no critical slowing down. We apply the algorithm in a
comparison of our perturbative results with Monte Carlo data.
