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 (Gn)→G then the gas process built on Gn 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 
lattices introduced in [Ann. Comb. 4 (2000) 269–284] and on a generalization of the 
lattices introduced in [J. Phys. A 29 (1996) 3357–3365].
Primary Subjects: 60K20, 60K35
Secondary Subjects: 68R05
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References
[1] Barcucci, E., Del Lungo, A., Pergola, E. and Pinzani, R. (1999). Directed animals, forests and permutations. Discrete Math. 204 41–71.
[2] Bétréma, J. and Penaud, J.-G. (1993). Animaux et arbres guingois. Theoret. Comput. Sci. 117 67–89. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
[3] Bétréma, J. and Penaud, J. G. (1993). Modèles avec particules dures, animaux dirigés, et séries en variables partiellement commutatives. Available at arXiv:math.CO/0106210.
[4] Bousquet-Mélou, M. (1998). New enumerative results on two-dimensional directed animals. Discrete Math. 180 73–106.
[5] Bousquet-Mélou, M. and Conway, A. R. (1996). Enumeration of directed animals on an infinite family of lattices. J. Phys. A 29 3357–3365.
[6] Corteel, S., Denise, A. and Gouyou-Beauchamps, D. (2000). Bijections for directed animals on infinite families of lattices. Ann. Comb. 4 269–284. Conference on Combinatorics and Physics (Los Alamos, NM, 1998).
[7] Dhar, D. (1983). Exact solution of a directed-site animals-enumeration problem in three dimensions. Phys. Rev. Lett. 51 853–856.
Mathematical Reviews (MathSciNet):
MR721768
[8] Gouyou-Beauchamps, D. and Viennot, G. (1988). Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem. Adv. in Appl. Math. 9 334–357.
Mathematical Reviews (MathSciNet):
MR956559
[9] Hakim, V. and Nadal, J. P. (1983). Exact results for 2D directed animals on a strip of finite width. J. Phys. A 16 L213–L218.
Mathematical Reviews (MathSciNet):
MR707373
[10] Le Borgne, Y. and Marckert, J.-F. (2007). Directed animals and gas models revisited. Electron. J. Combin. 14 R71 (electronic).
[11] Nadal, J. P., Derrida, B. and Vannimenus, J. (1982). Directed lattice animals in 2 dimensions: Numerical and exact results. J. Physique 43 1561–1574.
Mathematical Reviews (MathSciNet):
MR684783
[12] Viennot, G. (1985). Problèmes combinatoires posés par la physique statistique. Astérisque 121–122 225–246. Seminar Bourbaki, Vol. 1983/84.
Mathematical Reviews (MathSciNet):
MR768962
[13] Viennot, G. X. (1986). Heaps of pieces. I. Basic definitions and combinatorial lemmas. In Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985). Lecture Notes in Mathematics 1234 321–350. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR927773
