Publication : t03/086

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 $pm 1$. 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 $wp$ function with constrained periods. These results are used to analyze the survival probability of an evolving population spreading in the target spaces (i)-(ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.

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