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The functional RG (FRG) approach to pinning of d-dimensional manifolds is re-examined at any temperature T. The coupling function R(u) is shown to be a physical observable in any d, exactly related to a free energy cumulant in a parabolic well. In d = 0 its beta function is displayed to a high order, ambiguities resolved; for random field disorder (Sinai model) we obtain exactly the T = 0 fixed point R(u) and its thermal boundary layer (TBL) form (i.e. for u ~ T) at T > 0. Connection between FRG in d = 0 and decaying Burgers turbulence is discussed. An exact solution to the functional RG hierarchy in the TBL is obtained for any d and related to droplet probabilities.

Solutions of variable-length directed polymers are studied via a mapping onto the quantum mechanics of bosons with an external source in two dimensions. The theory is applicable, in particular, to entangled polymer nematics. We calculate the structure function of these systems, and the renormalized Frank constants.

We study numerically the (planar) graph colouring problem with q colours. For q = 4, when a perfect colouring can be achieved, the solutions are scattered "randomly" (as far as triangle correlations are concerned) in configuration space. On the contrary, for q = 3, colouring is always imperfect, but the optimal solutions seem to organize themselves in an ultrametric way. This could illustrate rather well the role of frustration on the configuration space landscape. We discuss the importance of the distance chosen.

It is suggested, on the basis of numerical simulations, that the (averaged) Lyapunov exponent at finite time of certain dynamical systems obeys universal scaling properties very similar to the scaling behaviour of the conductance in localization theory. An analytical approximation of the scaling function is derived from simple arguments. The possible mechanisms at the origin of this property are discussed, and the scaling function is proposed to be an important characterization of such dynamical systems.