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Combinatorial optimization is a fertile testing ground for
statistical physics methods developed in the
context of disordered systems, allowing
one to confront theoretical mean field predictions with actual
properties of finite dimensional systems. Our focus here is on minimum
matching problems, because they are computationally tractable while both
frustrated and disordered. We first study a mean field model taking the link
lengths between points to be independent random variables. For this model we
find perfect agreement with the results of a replica calculation, and
give a conjecture. Then we
study the case where the points to be matched are placed at random in a
d-dimensional Euclidean space.
Using the mean field model as an approximation to the Euclidean case,
we show numerically that the
mean field predictions are very accurate even at low dimension, and that the
error due to the approximation is O(1/d2). Furthermore,
it is possible to improve upon this approximation
by including the effects of Euclidean correlations among
k link lengths. Using k=3 (3-link correlations such as the triangle
inequality), the resulting errors in the energy density are already
less than 0.5% at d≥2. However, we argue that
the dimensional dependence of the Euclidean model's energy density
is non-perturbative, i.e., it is beyond all orders
in k of the expansion in k-link correlations
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