Two repulsive lines on disordered lattices

We investigate the ground-state properties of two lines with “on-site” repulsion on disordered Cayley tree and (Berker) hierarchical lattices, in connection with the question of multiple “pure states” for the corresponding one-line problem. Exact recursion relations for the distribution of ground-state energies and of the overlaps are derived. Based on a numerical study of the recursion relations, we establish that the total interaction energy on average is asymptotically proportional to the width δ of the ground-state energy fluctuation of a single line for both weak and strong (i.e., hard-core) repulsion. When the lengtht of the lines is finite, there is a finite probability of ordert −a for (nearly) degenerate, nonoverlapping one-line ground-state configurations, in which case the interaction energy vanishes. We show thata=ω (δ∼t ω) on hierarchical lattices. Monte Carlo transfer matrix calculation on a (1+1)-dimensional model yields the same scaling for the interaction energy but ana different from ω=1/3. Finitelength scalings of the distribution of the interaction energy and of the overlap are also discussed.