Self-avoiding tethered surfaces are always flat

The scaling behavior of fully flexible elastic tethered surfaces has been debated for decades. Some theories predict that self-avoiding surfaces would crumple in the absence of bending rigidity, while most simulations suggested that they would remain flat. Recent simulations on ideal membranes with lattice perforations suggest that systematically removing surface area from a membrane may provide an alternative way to crumpling self-avoiding surfaces. We perform extensive numerical simulations of two models of fully flexible elastic tethered surfaces in which self-avoidance can be systematically and continuously tuned to the ideal limit. We show that in the thermodynamic limit, these surfaces remain flat with a size exponent $\nu=1$ for any finite degree of self-avoidance, with or without membrane perforations.