Borodin, Alexei; Bufetov, Alexey; Olshanski, Grigori Limit shapes for growing extreme characters of \(U(\infty)\). (English) Zbl 1325.60013 Ann. Appl. Probab. 25, No. 4, 2339-2381 (2015). Summary: We prove the existence of a limit shape and give its explicit description for certain probability distributions on signatures (or highest weights for unitary groups). The distributions have representation theoretic origin – they encode the decomposition on irreducible characters of the restrictions of certain extreme characters of the infinite-dimensional unitary group \(U(\infty)\) to the growing finite-dimensional unitary subgroups \(U(N)\). The characters of \(U(\infty)\) are allowed to depend on \(N\). In a special case, this describes the hydrodynamic behavior for a family of random growth models in \((2+1)\)-dimensions with varied initial conditions. Cited in 12 Documents MSC: 60F05 Central limit and other weak theorems 60G70 Extreme value theory; extremal stochastic processes 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 22E66 Analysis on and representations of infinite-dimensional Lie groups Keywords:infinite-dimensional unitary group; limit shapes; probability distributions; extreme characters; signature PDFBibTeX XMLCite \textit{A. Borodin} et al., Ann. Appl. Probab. 25, No. 4, 2339--2381 (2015; Zbl 1325.60013) Full Text: DOI arXiv Euclid References: [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118 . Cambridge Univ. Press, Cambridge. · Zbl 1184.15023 [2] Biane, P. (1995). Representations of unitary groups and free convolution. Publ. Res. Inst. Math. Sci. 31 63-79. · Zbl 0856.22017 [3] Biane, P. (2001). Approximate factorization and concentration for characters of symmetric groups. Int. Math. Res. 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