  Consider the problem of matching two independent i.i.d. samples of size $N$
from two distributions $P$ and $Q$ in $\mathbb{R}^d$. For an arbitrary
continuous cost function, the optimal assignment problem looks for the matching
that minimizes the total cost. We consider instead in this paper the problem
where each matching is endowed with a Gibbs probability weight proportional to
the exponential of the negative total cost of that matching. Viewing each
matching as a joint distribution with $N$ atoms, we then take a convex
combination with respect to the above Gibbs probability measure. We show that
this resulting random joint distribution converges, as $N\rightarrow \infty$,
to the solution of a variational problem, introduced by F\"ollmer, called the
Schr\"odinger problem. We also derive the first two error terms of orders
$N^{-1/2}$ and $N^{-1}$, respectively. This gives us central limit theorems for
integrated test functions, including for the cost of transport, and second
order Gaussian chaos limits when the limiting Gaussian variance is zero. The
proofs are based on a novel chaos decomposition of the discrete Schr\"odinger
bridge by polynomial functions of the pair of empirical distributions as the
first and second order Taylor approximations in the space of measures. This is
achieved by extending the Hoeffding decomposition from the classical theory of
U-statistics.
