{ "id": "cond-mat/0306602", "version": "v2", "published": "2003-06-24T12:52:28.000Z", "updated": "2003-10-17T13:16:51.000Z", "title": "Random trees between two walls: Exact partition function", "authors": [ "J. Bouttier", "P. Di Francesco", "E. Guitter" ], "comment": "25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusion", "journal": "J. Phys. A: Math. Gen. 36 (2003) 12349-12366", "doi": "10.1088/0305-4470/36/50/001", "categories": [ "cond-mat.stat-mech", "math.CO", "nlin.SI" ], "abstract": "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.", "revisions": [ { "version": "v2", "updated": "2003-10-17T13:16:51.000Z" } ], "analyses": { "keywords": [ "exact partition function", "random trees", "target space", "non-trivial partition function", "planar graphs" ], "tags": [ "journal article" ], "note": { "typesetting": "TeX", "pages": 25, "language": "en", "license": "arXiv", "status": "editable" } } }