  We use Monte Carlo simulations to measure the spin-spin correlation function
in the disordered phase of two-dimensional $q$-state Potts models with
$q=10,15$, and $20$ at the first-order transition point $\beta_t$. To extract
the correlation length $\xi_d$ from the exponential decay of the correlation
function over several decades with the desired accuracy we make extensively use
of cluster-update techniques and improved estimators. Our results for $\xi_d$
are compatible with an analytic formula. As a byproduct we also measure the
energy moments in the disordered phase and find very good agreement with a
recent large $q$ expansion at $\beta_t$.
