AbstractWe 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
.