  In this thesis, we investigate the asymptotics of random partitions chosen
according to probability measures coming from the representation theory of the
symmetric groups $S_n$ and of the finite Chevalley groups $GL(n,F_q)$ and
$Sp(2n,F_q)$. More precisely, we prove laws of large numbers and central limit
theorems for the $q$-Plancherel measures of type A and B, the Schur-Weyl
measures and the Gelfand measures. Using the RSK algorithm, it also gives
results on longest increasing subsequences in random words. We develop a
technique of moments (and cumulants) for random partitions, thereby using the
polynomial functions on Young diagrams in the sense of Kerov and Olshanski. The
algebra of polynomial functions, or observables of Young diagrams is isomorphic
to the algebra of partial permutations; in the last part of the thesis, we try
to generalize this beautiful construction.
