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October 2009 A note on the enumeration of directed animals via gas considerations
Marie Albenque
Ann. Appl. Probab. 19(5): 1860-1879 (October 2009). DOI: 10.1214/08-AAP580

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 (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 $\mathcal{T}_{n}$ lattices introduced in [Ann. Comb. 4 (2000) 269–284] and on a generalization of the $\mathcal{L}_{n}$ lattices introduced in [J. Phys. A 29 (1996) 3357–3365].

Citation

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Marie Albenque. "A note on the enumeration of directed animals via gas considerations." Ann. Appl. Probab. 19 (5) 1860 - 1879, October 2009. https://doi.org/10.1214/08-AAP580

Information

Published: October 2009
First available in Project Euclid: 16 October 2009

zbMATH: 1194.60056
MathSciNet: MR2569810
Digital Object Identifier: 10.1214/08-AAP580

Subjects:
Primary: 60K20 , 60K35
Secondary: 68R05

Keywords: cyclic Markov chains , directed animals , Gas model , graphs convergence

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 5 • October 2009
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