  We establish some exact asymptotic results for a matching problem with
respect to a family of beta distributions. Let $X_1, \ldots, X_n$ be
independent random variables with common distribution the symmetric Jacobi
measure $d\mu (x) = C_d (1-x^2)^{\frac d2 -1} dx$ with dimension $ d \geq 1$ on
$[-1, 1]$, and let $\mu_n = \frac{1}{n} \sum_{i = 1}^{n} \delta_{X_i}$ be the
associated empirical measure. We show that $\lim_{n \to \infty} n\E \left[
W_2^2( \mu^n, \mu ) \right] = \sum_{k = 1}^{\infty} \frac{1}{k(k+d-1)}$, where
$W_2$ is the quadratic Kantorovich distance with respect to the intrinsic cost
$\rho(x, y) = |\arccos(x) - \arccos (y)|$, $(x, y) \in [-1, 1]^2$, associated
to the model. When $\mu$ is the product measure of two Jacobi measures with
dimensions $d$ and $d'$ respectively, then $\E \left[ W_2^2( \mu^n, \mu )
\right] \approx \frac{\log n}{n}$. In the particular case $d = d' = 1$
(corresponding to the product of arcsine laws), $\lim_{n \to \infty}
\frac{n}{\log n} \E \left[ W_2^2( \mu^n, \mu ) \right] = \frac{\pi}{4}$.
Similar results do hold for non-symmetric Jacobi distributions. The proofs are
based on the recent PDE and mass transportation approach developed by
L.~Ambrosio, F.~Stra and D.~Trevisan.
