Determination of the Crumpling Fractal Dimension via k-space MCRG

We have to go back to the late eighties[1,2] to find the first evidence for the existence of a phase transition separating the crumpled and smooth regimes in mathematical models of surfaces defined on a mesh with fixed connectivity. However, the actual values of the critical exponents associated to this phase transition remain largely unknown. Their determination is crucial to identify which type of conformal theory one gets in the vicinity of the critical point. Two recent numerical simulations report the following values for the critical exponents α and ν. In [3] a direct measuremt of the correlation length gives ν = 0.71 ± 0.05 and α = 0.58 ± 0.1 (the latter is obtained via scaling relations). A direct measurement of the finite size scaling of the specific heat gives ν = 0.73 ± 0.06, in agreement with the previous values. In [4] finite size scaling gives ω = α/ν = 1.11±0.1, that is α = 0.71±0.05, ν = 0.64 ± 0.02. However these values do not agree with a direct fit to the specific heat results on large lattices (up to 128), which tend to give much lower values for α [5,6]. Virtually nothing is known of the other critical exponents η and γ, which cannot be deduced from ν and α alone using scaling relations. The most interesting one is perhaps η, the anomalous dimension of the field xi that describes the position of the surface. The fractal dimension dH is