On the Convergence of Cluster Expansions for Polymer Gases |
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Authors: | Rodrigo Bissacot Roberto Fernández Aldo Procacci |
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Affiliation: | 1.Departamento de Matemática-ICEx,UFMG,Belo Horizonte,Brazil;2.Laboratoire de Maths Raphael Salem,Université de Rouen,Rouen,France;3.Department of Mathematics,Utrecht University,Utrecht,The Netherlands |
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Abstract: | We compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard-core exclusions, with emphasis on polymers defined as finite subsets of a countable set (e.g. contour expansions and more generally high- and low-temperature expansions). In order of increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion. We show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of Dobrushin inductive argument and by an alternative—in fact, more elementary—handling of the Kirkwood-Salzburg equations. In addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive Dobrushin argument and, furthermore, to a systematic way to improve bounds on correlations. |
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