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A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers.
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A. Borodin and G. Olshanski, Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. 161 (2005), no. 3, 1319–1422, arXiv:math/0109194
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P. Sniady, Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probability Theory and Related Fields, 136, no. 2 (2006), 263–297
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2017
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