2010/09/21 by Pierre‐Loïc Méliot, Pierre-Loïc Méliot, Méliot, Pierre-Loïc
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Random Matrices and Applications #Representation Theory (math.RT) #Spectral Theory in Mathematical Physics #math.RT
paper · pdf · doi:10.48550/arxiv.1009.4047
21 pages, 8 figures
arxiv created 2010/09/21 · openalex publication_date 2010/09/21 · arxiv updated 2010/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If a partition λ of size n is chosen randomly according to the Plancherel measure Pn[λ] = (dim λ)2/n!, then as n goes to infinity, the rescaled shape of λ is with high probability very close to a non-random continuous curve Ω known as the Logan-Shepp-Kerov-Vershik curve. Moreover, the rescaled deviation of λ from this limit shape can be described by an explicit generalized gaussian process. In this paper, we investigate the analoguous problem when λ is chosen with probability proportional to dim λ instead of (dim λ)2. We shall use very general arguments due to Ivanov and Olshanski for the first and second order asymptotics (cf. arXiv:math/0304010); these arguments amount essentially to a method of moments in a noncommutative setting. The first order asymptotics of the Gelfand measures turns out to be the same as for the Plancherel measure; on the contrary, the fluctuations are different (and bigger), although they involve the same generalized gaussian process. Many of our computations relie on the enumeration of involutions and square roots in Sn.