Overlap properties and adsorption transition of two Hamiltonian paths

We consider a model of two (fully) compact polymer chains, coupled through an attractive interaction. These compact chains are represented by Hamilton ian paths (HP), and the coupling favors the existence of common bonds betwe en the chains. We use a (n = 0 component) spin representation for these pat hs, and we evaluate the resulting partition function within a homogeneous s addle point approximation. For strong coupling (i.e. at low temperature), o ne finds a phase transition towards a "frozen" phase where one chain is com pletely adsorbed onto the other. By performing a Legendre transform, we obt ain the probability distribution of overlaps. The fraction of common bonds between two HP, i.e. their overlap q, has both lower (q(m)) and upper (q(M) ) bounds. This means in particular that two HP with overlap greater than q( M) coincide. These results may be of interest in (bio)polymers and in optim ization problems.