Combinatorics of bicubic maps with hard particles

J. Bouttier, P. Di Francesco, E. Guitter

Research output: Contribution to journalArticlepeer-review


We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by the use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of 'admissible' trees with possibly doubly occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.

Original languageEnglish (US)
Pages (from-to)4529-4559
Number of pages31
JournalJournal of Physics A: Mathematical and General
Issue number21
StatePublished - May 27 2005
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)


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