  In this paper, we study a class of multilinear Gibbs measures with
Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general
base measure. Expressing the asymptotic free energy as an optimization problem
over a space of functions, we obtain necessary and sufficient conditions for
replica-symmetry. Utilizing this, we obtain weak limits for a large class of
statistics of interest, which includes the ''local fields/magnetization'', the
Hamiltonian, the global magnetization, etc. An interesting consequence is a
universal weak law for contrasts under replica symmetry, namely,
$n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our
results yield a probabilistic interpretation for the optimizers arising out of
the limiting free energy. We also prove the existence of a sharp phase
transition point in terms of the temperature parameter, thereby generalizing
existing results that were only known for quadratic Hamiltonians. As a
by-product of our proof technique, we obtain exponential concentration bounds
on local and global magnetizations, which are of independent interest.
