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Random trees between two walls: Exact partition function | J. Bouttier
; P. Di Francesco
; E. Guitter
; | Date: |
24 Jun 2003 | Journal: | J. Phys. A: Math. Gen. 36 (2003) 12349-12366 | Subject: | Statistical Mechanics; Combinatorics; Exactly Solvable and Integrable Systems | 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. | Source: | arXiv, cond-mat/0306602 | Services: | Forum | Review | PDF | Favorites |
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