A NOTE ON THE ENUMERATION OF DIRECTEDANIMALS VIA GAS CONSIDERATIONSBy Marie

arXiv: math.PR/0805.1349 A NOTE ON THE ENUMERATION OF DIRECTED ANIMALS VIA GAS CONSIDERATIONS By Marie Albenque Université Bordeaux 1 and Université Paris 7 In the literature, most of the results about the enumeration of dire ted animals on latti es via gas onsiderations are obtained by a formal passage to the limit of enumeration of dire ted animals on y li al versions of the latti e. Here we provide a new point of view on this phenomenon. Using the gas onstru tion given in [10℄, we des ribe the gas pro ess on the y li al versions of the latti es as a y li al Markov hain (roughly speaking, Markov hains onditioned to ome ba k to their starting point). Then we introdu e a notion of onvergen e of graphs, su h that if (Gn) → G then the gas pro ess built on Gn onverges in distribution to the gas pro ess on G. That gives a general tool to show that gas pro esses related to animals enumeration are often Markovian on lines extra ted from latti es. We provide examples and omputations of new generating fun tions for dire ted animals with various sour es on the triangular latti e, on the Tn latti es introdu ed in [6℄ and on a generalization of the Ln latti es introdu ed in [5℄. 1. Introdu tion. Let G = (V,E) be a dire ted graph with set of verti es V and set of oriented edges E. Let A and S be two subsets of V , with S ⊂ A. We say that A is a dire ted animal (DA) with sour e S if and only if any vertex of A an be rea hed from an element of S through a dire ted path having all its verti es in A (see Figure 1). The verti es of A are alled ells and the number of ells, denoted |A|, is the area of A. We denote GG S the generating fun tion (GF) for DA on G with sour e S ounted a ording to their area: G S (t) = ∑