Random trees between two walls: Exact partition function

TL;DR

This paper derives the exact partition function for a model of random trees constrained between walls, revealing a soliton-like structure and continuum limits involving elliptic functions, with applications to population spread and planar graphs.

Contribution

It provides the first exact solution for bounded random trees with wall constraints, including their continuum scaling limits and applications.

Findings

Exact partition function derived for bounded random trees.

Continuum limit expressed via Weierstrass p-function.

Applications to population dynamics and planar graph enumeration.

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.