Tethered membranes with long-range interactions

We treat the problem of D-dimensional tethered (or polymerized) membranes with long-range repulsive interactions (varying as 1/r^γ) in a d-dimensional embedding space using both large-d and variational methods. We find three different regimes in the ({γ},D) plane: for small D, the manifold is always crumpled, either Gaussian or swollen (with the exact exponent for the radius of gyration {ν}=2D/{γ}); for intermediate D, it undergoes a crumpling transition; and for large D, it is always flat. By extrapolating these results to the case of short-range interactions, we propose a phase diagram in the (d,D) plane for the still poorly understood problem of self-avoiding manifolds. The physical point D=2 and d=3 lies at the boundary of the always-flat region. This may provide an explanation for the recent numerical simulations that seemingly show that self-avoidance always makes two-dimensional membranes flat.