The Number of Optimal Matchings for Euclidean Assignment on the Line

Abstract

We consider the Random Euclidean Assignment Problem in dimension $$d=1$$ 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, $$\sim \exp (S_N)$$ ∼ 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 $$S_N$$ S N (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large N, $$S_N \sim \frac{1}{2} N \log N + N s + {\mathcal {O}}\left( \log N \right) $$ S N ∼ 1 2 N log N + N s + O log N , 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 $$S_N$$ 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 $$S_N$$ S N .

Minimal matchings of point processes

“…See for example [1,4,5,8,15,18,29,31]. We mention in particular the recent remarkably precise analysis of 2-minimal matchings in dimension 2 in [2,3], and, in dimension 1, the study of finite γ -minimal matchings for γ < 0, and of the number of finite 1-minimal matchings in [7] and [9] respectively. The equivalence of all γ > 1 (and indeed all convex cost functions) in d = 1 (see Theorem 3(i)) has been observed previously in finite settings; see for example [30,Ch.…”