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In this paper, we review the representation theory of the infinite symmetricgroup, and we extend the works of Kerov and Vershik by proving that theirreducible characters of the infinite symmetric group always satisfy a centrallimit theorem. Hence, for any point of the Thoma simplex, the correspondingmeasures on the levels of the Young graph have a property of gaussianconcentration. By using the Robinson-Schensted-Knuth algorithm and the theoryof Pitman operators, we relate these results to the properties of certainrandom permutations obtained by riffle shuffles, and to the behaviour of randomwalks conditioned to stay in a Weyl chamber.

A central limit theorem for the characters of the infinite symmetric group and of the infinite Hecke algebra