论文标题
稀疏相互作用图的两体分区功能的森林扩展
Forest expansion of two-body partition functions for sparse interaction graphs
论文作者
论文摘要
我们通过Brydges-Kennedy-Abdessalam-Rivasseau森林扩张研究了对经典两体分区功能的树近似值。我们表明，对于稀疏图（带有大周期），可以通过图形多项式扩展在相互作用图的森林上近似一定温度$ t^*$的分区函数。在此“森林阶段”中，我们表明近似值可以用参考树$ \ natercal t $在交互图上写成，并由于周期而进行校正。从这个角度来看，这意味着高温模型易于在稀疏图上求解，因为人们可以使用信念传播评估分区函数。我们还表明，存在一个高温和低温制度，其中可以通过（给定的）加权图上的最大跨越树算法获得$ \ Mathcal t $。我们研究这些校正的代数，并向树Ansatz提供一阶和二阶近似，并为一阶近似提供明确的示例。
We study tree approximations to classical two-body partition functions on sparse and loopy graphs via the Brydges-Kennedy-Abdessalam-Rivasseau forest expansion. We show that for sparse graphs (with large cycles), the partition function above a certain temperature $T^*$ can be approximated by a graph polynomial expansion over forests of the interaction graph. Within this "forest phase", we show that the approximation can be written in terms of a reference tree $\mathcal T$ on the interaction graph, with corrections due to cycles. From this point of view, this implies that high-temperature models are easy to solve on sparse graphs, as one can evaluate the partition function using belief propagation. We also show that there exists a high- and low-temperature regime, in which $\mathcal T$ can be obtained via a maximal spanning tree algorithm on a (given) weighted graph. We study the algebra of these corrections and provide first- and second-order approximation to the tree Ansatz, and give explicit examples for the first-order approximation.
