Polymer Collapse on Fluctuating Random Surfaces

The conformations of interacting linear polymers on a dynamical planar random lattice are studied using a random two-matrix model. An exact expression for the partition function of self-avoiding chains subject to attractive contact interactions of relative strength $1/c$, $0<c<1$, is found as a function of chain length $L$ and $c$. The number of configurations $L^a {\rm e}^{bL}$ as $L\to \infty$ is determined, showing that a chain undergoes a third-order collapse transition at $c=\sqrt{2} -1$; the universal scaling laws are found.