Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by 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.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde