Gaussian fluctuations of characters of symmetric groups and of Young diagrams
arXiv:math/0501112 · doi:10.1007/s00440-005-0483-y
Abstract
We study asymptotics of reducible representations of the symmetric groups S_q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, left-regular representation (Plancherel measure), tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.
37 pages; version 3: conceptual change in the proofs
References in corpus (2)
Cited by in corpus (10)
- Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants
- Exponential Approximation by Stein's Method and Spectral Graph Theory
- Asymptotics of characters of symmetric groups, genus expansion and free probability
- Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
- Representations of Lie groups and random matrices
- Zero biasing and growth processes
- A central limit theorem for singular graphons
- Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group
- Modulus of continuity of Kerov transition measure for continual Young diagrams
- Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures