  In this paper, we review the representation theory of the infinite symmetric
group, and we extend the works of Kerov and Vershik by proving that the
irreducible characters of the infinite symmetric group always satisfy a central
limit theorem. Hence, for any point of the Thoma simplex, the corresponding
measures on the levels of the Young graph have a property of gaussian
concentration. By using the Robinson-Schensted-Knuth algorithm and the theory
of Pitman operators, we relate these results to the properties of certain
random permutations obtained by riffle shuffles, and to the behaviour of random
walks conditioned to stay in a Weyl chamber.
