On directed interacting animals and directed percolation

TL;DR

This paper investigates the phase diagram of directed lattice animals with interactions, revealing their collapse transition aligns with directed percolation, and provides precise critical exponents through renormalization group analysis.

Contribution

It introduces a detailed analysis of directed lattice animals, connecting their collapse transition to directed percolation and calculating critical exponents with a phenomenological renormalization approach.

Findings

Collapse transition is equivalent to directed bond percolation threshold.

Critical exponents for animal size and fugacities are estimated.

Crossover exponent equals the Fisher exponent for directed percolation.

Abstract

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.