We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called height functions, and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the ‘length’ of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.

中文翻译：

---

加权自我规避步行

我们研究了无限图和组上加权自我规避步行（SAW）的结缔常数。主要关注于有限生成的，几乎可指示的组上的加权SAW。这样的组具有所谓的高度函数，这允许研究具有桥梁特殊性的声表面波。组结构与高度函数和权重函数之间的交互作用有关。当权重函数的支持不受限制时会出现主要困难，因为相应的图不再是局部有限的。有两个主要结果，其中第一个是加权连接常数和加权桥常数相等的条件。当权重函数得到无穷的支持时，我们将采用步行“长度”的广义概念，这要符合一定的条件。在第二个主要结果中，上述等式用于证明加权函数具有适当距离函数的空间上的连接常数的连续性定理。