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Article Dans Une Revue Journal of Physics A: Mathematical and General (1975 - 2006) Année : 2003

Random trees between two walls: exact partition function

Résumé

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.

Dates et versions

hal-00586660 , version 1 (18-04-2011)

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Jérémie Bouttier, Philippe Di Francesco, Emmanuel Guitter. Random trees between two walls: exact partition function. Journal of Physics A: Mathematical and General (1975 - 2006), 2003, 36 (50), pp.12349 - 12366. ⟨10.1088/0305-4470/36/50/001⟩. ⟨hal-00586660⟩
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