度有界图的左右收敛性
Left and right convergence of graphs with bounded degree
摘要：
在稠密图和有界度图这两种极端情况下，给出了收敛图序列的理论。我们可以通过计算从固定图到序列成员的同态（左收敛）或计算从固定图到序列成员的同态（右收敛）来定义收敛。在适当的条件下，Borgs、Chayes、Lovász、Sós和Vesztergombi证明了这两种定义收敛的方法在稠密情形下是等价的。本文在有界度情形下建立了一个类似的等价关系。在统计物理学方面，左收敛蕴涵右收敛意味着对于左收敛序列，一大类统计物理模型的分拆函数收敛。这个证明依赖于统计物理学的技术，如集群扩展和Dobrushin Uniform。
英文摘要：
The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovász, Sós and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left-convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness.
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