Abstract
We give a noncommutative extension of Kerov’s central limit theorem for irreducible characters of the symmetric group with respect to the Plancherel measure [S.Kerov:C.R.Acad.Sci.Paris 316 (1993)]in the framework of algebraic probability theory.For adjacency operators associated with the cycle classes, we consider their decomposition according to the length function on the Cayley graph of the symmetric group.We develop a certain noncommutative central limit theorem for them, in which the limit picture is described by creation and annihilation operators on an analogue of the Fock space equipped with an orthonormal basis labelled by Young diagrams.The limit Gaussian measure in Kerov’s theorem appears as the spectral distribution of the field operators in our setting.
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References
Accardi, L., Bożejko, M.: Interacting Fock spaces and Gaussianization of probability measures. Infin.Dimen.Anal.Quantum Probab.Relat.Top.,1, 663–670(1998)
Bannai, E., Ito, T.: Algebraic combinatorics I,association schemes. Menlo Park, California, Benjamin /Cummings (1984)
Biane, P.: Permutation model for semi-circular systems and quantum random walks.Pacific J.Math.,171, 373–387 (1995)
Biane, P.:Representations of symmetric groups and free probability. Advances in Math.,138, 126–181 (1998)
Hashimoto, Y.: Deformations of the semicircular law derived from random walks on free groups.Probab.Math.Stat.,18, 399–410 (1998)
Hashimoto, Y.: Quantum decomposition in discrete groups and interacting Fock spaces.Infin.Dimen.Anal.Quantum Probab.Relat.Top.,4, 277–287 (2001)
Hashimto, Y., Hora, A., Obata, N.:Central limit theorems for large graphs:a method of quantum decomposition.Preprint (2001)
Hashimoto, Y., Obata, N., Tabei, N.:A quantum aspect of asymptotic spectral analysis of large Hamming graphs.In: Hida, T., Saitô, K.(eds) QuantumInformation III, Singapore, World Scientific, 45–57 (2001)
Hora, A.: Central limit theorem for the adjacency operators on the infinite symmetric group.Commun.Math.Phys.,195, 405–416 (1998)
Hora, A.: Central limit theorems and asymptotic spectral analysis on large graphs.Infin.Dimen.Anal.Quantum Probab.Relat.Top.,1, 221–246 (1998)
Hora, A.:Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians.Probab.Theory Relat. Fields,118, 115–130 (2000)
Ivanov, V., Olshanski, G.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams.Preprint,(2001)
Kerov, S.: Gaussian limit for the Plancherel measure of the symmetric group.C.R.Acad.Sci.Paris,316, Série I, 303–308 (1993)
Logan, B.F., Shepp, L.A.: A variational problem for random Young tableaux.Advances in Math.,26, 206–222 (1977)
Vershik, A.M., Kerov, S.V.: Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux.Doklady AN SSSR, 233, 1024–1027 (1977);English translation:Soviet Mathematics Doklady,18,527-531 (1977)
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Hora, A. (2003). A Noncommutative Version of Kerov’s Gaussian Limit for the Plancherel Measure of the Symmetric Group. In: Vershik, A.M., Yakubovich, Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics(), vol 1815. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44890-X_4
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DOI: https://doi.org/10.1007/3-540-44890-X_4
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