Two repulsive lines on disordered lattices

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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.
Original languageEnglish
Pages (from-to)581–606
Number of pages26
JournalJournal of Statistical Physics
Issue number3-4
Publication statusPublished - Nov 1994

User-Defined Keywords

  • Directed polymer
  • disorder
  • hierarchical lattice
  • overlap
  • rare event
  • replica


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