Scaling theory for the size of crumbled membranes in presence of linear polymers and other objects

Condensed systems of polymers and membranes are studied. First it is shown that melts of membranes behave quite differently to melts of linear polymers. Whereas polymer chains in melts exhibit random walk behaviour, surfaces are always saturated. Most striking are results where surfaces are dissolved in linear polymers, where the size of th membrane is not changed, i.e. $\nu = 4/5$ as in the crumbled phase. The general value of $\nu$ for $D$-dimensional manifolds is given by $\nu = 2D/(2+d)$ where in the crumbled phase $\nu =({2+D})/({2+d})$. Both values are identical for $D=2$. It is further shown that even for mixtures of membranes and stiff objects (Tobacco mosaic virusses) the crumbling exponent is also $\nu = 4/5$. This is a speciality for $D=2$ membranes. For general values of $D$ new exponents are predicted