Two-dimensional filling in ordered and disordered systems

We consider filling or wedge-wetting transitions occurring in a (1 + 1)-dimensional wedge geometry with both thermal and random-bond disorder using effective interfacial Hamiltonian models. For ordered systems the problem may be solved using transfer-matrix methods for quite arbitrary choices of interfacial binding potential and gives a complete classification of the possible critical behaviours. For random bonds the transition is studied for short-ranged forces using the replica trick and the wedge-wetting critical exponents are identified. Our results establish a remarkable relation between the mid-point height probability distribution PF(l0) at filling transitions and the appropriately defined height distribution function Pπ(l;θπ) at planar wetting transitions. We observe that provided the wetting specific heat component αs = 0, then, in the scaling limit, PF(l0) = Pπ(l;θπ-α) where θπ is the contact angle and α is the tilt angle of the wedge. This relation completely determines the allowed values of the filling critical exponents in the fluctuation-dominated regimes. Conjectures regarding interfacial fluctuation effects in finite-size two-dimensional systems are also made.