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.
For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High T Convergence
设Psi_0为量子海森堡铁磁体sharp i sigma_zi的波函数，Psi_mu = exp（-mu*H）Psi_0.我们研究的期望类似的形式<Psi_mu，（sigma_zi）Psi_mu>/<Psi_mu，Psi_mu>，我们提出了一个正式的聚合物扩展，其收敛性，我们证明了足够小的亩。本文的方法是将波函数Psi_mu与它的一个近似，即乘积函数相关联。在乘积近似的第j个点上，上分量是phi_mu（j），下分量是（1-phi_mu（j）），其中phi满足晶格热方程。这通过簇或聚合物展开来显示。目前的工作开始在以前的文件，主要是一个数值研究，并提供了一个证明的结果有关的猜想3这以前的文件。
Heat flow in anharmonic crystals with internal and external stochastic baths: A convergent polymer expansion for a model with discrete time and long range interparticle interaction
