A note on the enumeration of directed animals via gas considerations

TL;DR

This paper introduces a new approach to enumerating directed animals on lattices using gas processes modeled as cyclical Markov chains, providing new generating functions and convergence tools.

Contribution

It offers a novel perspective on gas processes for directed animal enumeration, connecting them to cyclical Markov chains and graph convergence.

Findings

Gas processes on cyclical lattices are Markovian on certain lines.

New generating functions for directed animals on various lattices.

A general convergence framework for gas processes on graphs.

Abstract

In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. Here we provide a new point of view on this phenomenon. Using the gas construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we introduce a notion of convergence of graphs, such that if $(G_n)\to G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. That gives a general tool to show that gas processes related to animals enumeration are often Markovian on lines extracted from lattices. We provide examples and computations of new generating functions for directed animals with various sources on the triangular lattice, on the $\mathcal {T}_n$ lattices introduced in [Ann. Comb. 4 (2000) 269--284] and on a generalization of the $\mathcaligr {L}_n$ lattices introduced in [J. Phys. A 29 (1996) 3357--3365].