Mathematics – Probability
Scientific paper
2008-05-09
Annals of Applied Probability 2009, Vol. 19, No. 5, 1860-1879
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/08-AAP580 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/08-AAP580
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].
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