Abstract. We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary order. They can, however, be broken down completely to products of pair functions depending on two time arguments only. For binary couplings J = ±I a spin glass solution is found which approaches the corresponding solution for the SK-model in the limit of high connectivity. For more general distributions P (J) no stable or marginal solution of this type appears to exist. The nature of the low temperature phase in this more general case is unclear.PACS numbers: 05.20.-y(Classical statistical mechanics), 75.10.Nr (Spin glass and other random models)

Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semi-microscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long range orientational order or are frozen in random directions depending on the value of the crossing angle, the crosslink concentration, and the stiffness of the polymers. A crossing angle θ ∼ 2π/M leads to long range M -fold orientational order, e.g., "hexatic" or "tetratic" for θ = 60 • or 90 • , respectively. The transition to the orientationally ordered state is discontinuous and the critical cross-link density, which is higher than that of the gelation transition, depends on the bending stiffness of the polymers and the crosslink angle: the higher the stiffness and the lower M , the lower the critical number of cross-links. In between the sol and the long range ordered state, we always expect a gel which is a statistically isotropic amorphous solid (SIAS) with random positional and random orientational localization of the participating polymers.