随机欧几里德匹配问题中的单圈图
One-loop diagrams in the Random Euclidean Matching Problem
摘要：
匹配问题是一个臭名昭著的组合优化问题，多年来一直吸引着统计物理学界的关注。在这里，我们分析的欧几里德版本的问题，即最佳匹配问题之间的点随机分布在一个$d$维欧几里德空间，其中的成本最小化取决于点的成对距离。使用迈耶的集群扩展，我们写了一个正式的表达式的复制的行动，是适合鞍点计算。我们给出了每项展开式的图解规则，并详细分析了单回路图。理论的一个特点是，当图形周围的平均场部分的行动微扰计算，是零动量的质量消失。在非欧几里德的情况下，不相关的成本，而不是，我们预测和数值验证一个异常缩放的子子领先的校正渐近平均成本。
英文摘要：
The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e. the optimal matching problem between points randomly distributed on a $d$-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.
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