2003/06/30 by J. Bouttier, Jérémie Bouttier, Philippe Di Francesco +3
Computer Science · Mathematics · Physics and Astronomy · #Algorithms and Data Compression #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #cond-mat.stat-mech #math.CO #nlin.SI
paper · pdf · doi:10.1088/0305-4470/36/50/001
published as J. Phys. A: Math. Gen. 36 (2003) 12349-12366 · 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusion
arxiv created 2003/10/17 · openalex publication_date 2003/12/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labelled by integers representing their position in the target space, with the solid-on-solid constraint that adjacent vertices have labels differing by ±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 ℘ function with constrained periods. These results are used to analyse 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.