We consider the Random Euclidean Assignment Problem in dimension \(d=1\), with linear cost function. In this version of the problem, in general, there is a large degeneracy of the ground state, i.e. there are many different optimal matchings (say, \(\sim \exp (S_N)\) at size N). We characterize all possible optimal matchings of a given instance of the problem, and we give a simple product formula for their number. Then, we study the probability distribution of \(S_N\) (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large N, \(S_N \sim \frac{1}{2} N \log N + N s + {\mathcal {O}}\left( \log N \right) \), where s is a random variable whose distribution p(s) does not depend on N. We give expressions for the moments of p(s), both from a formulation as a Brownian process, and via singularity analysis of the generating functions associated to \(S_N\). The latter approach provides a combinatorial framework that allows to compute an asymptotic expansion to arbitrary order in 1/N for the mean and the variance of \(S_N\).

我们考虑具有线性成本函数的维度\（d = 1 \）中的随机欧几里得分配问题。通常，在此问题的版本中，基态存在很大的退化，即，存在许多不同的最佳匹配（例如，大小为N的\（\ sim \ exp（S_N）\））。我们描述了给定问题实例的所有可能最佳匹配，并给出了其数量的简单乘积公式。然后，我们研究均匀随机集合中\（S_N \）（模型的零温度熵）的概率分布。我们发现，对于大N，\（S_N \ sim \ frac {1} {2} N \ log N + N s + {\数学{O}} \ left（\ log N \ right）\），其中s是随机变量，它的分布p（小号）不依赖于Ñ。我们给出了p（s）时刻的表达式，既可以通过将其表示为布朗过程，也可以通过对与\（S_N \）相关的生成函数的奇异性分析来给出。后一种方法提供了一个组合框架，该框架允许针对\（S_N \）的均值和方差，以1 / N的形式渐近展开至任意阶数。