×

zbMATH — the first resource for mathematics

Asymptotics of Plancherel measures for the infinite-dimensional unitary group. (English) Zbl 1153.60058
Summary: We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.
We show that any measure from our family defines a determinantal point process on \(\mathbb Z_+ \times \mathbb Z\), and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.

MSC:
60K40 Other physical applications of random processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv Link
References:
[1] Aptekarev, A.; Bleher, P.; Kuijlaars, A., Large n limit of Gaussian random matrices with external source, part II, Comm. math. phys., 259, 2, 367-389, (2005) · Zbl 1129.82014
[2] Baik, J.; Deift, P.; Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations, J. amer. math. soc., 12, 4, 1119-1178, (1999) · Zbl 0932.05001
[3] Baik, J.; Deift, P.; Johansson, K., On the distribution of the length of the second row of a Young diagram under Plancherel measure, Geom. funct. anal., 10, 4, 702-731, (2000) · Zbl 0963.05133
[4] Biane, P., Approximate factorization and concentration for characters of symmetric groups, Int. math. res. not., 2001, 4, 179-192, (2001) · Zbl 1106.20304
[5] Borodin, A., Periodic Schur process and cylindric partitions, Duke math. J., 140, 3, 391-468, (2007) · Zbl 1131.22003
[6] Borodin, A.; Ferrari, P., Large time asymptotics of growth models on space-like paths I: pushasep
[7] Borodin, A.; Ferrari, P.; Prähofer, M.; Sasamoto, T., Fluctuation properties of the TASEP with periodic initial configuration, J. stat. phys., 129, 1055-1080, (2007) · Zbl 1136.82028
[8] Borodin, A.; Okounkov, A.; Olshanski, G., Asymptotics of Plancherel measures for symmetric groups, J. amer. math. soc., 13, 3, 481-515, (2000) · Zbl 0938.05061
[9] Borodin, A.; Olshanski, G., Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of math. (2), 161, 3, 1319-1422, (2005) · Zbl 1082.43003
[10] Borodin, A.; Olshanski, G., Stochastic dynamics related to Plancherel measure on partitions, Amer. math. soc. transl. ser. 2, 217, 9-22, (2006) · Zbl 1109.60041
[11] Borodin, A.; Olshanski, G., Asymptotics of Plancherel-type random partitions, J. algebra, 313, 1, 40-60, (2007) · Zbl 1117.60051
[12] Borodin, A.; Rains, E., Eynard-mehta theorem, Schur process, and their Pfaffian analogs, J. stat. phys., 121, 3-4, 291-317, (Nov. 2005)
[13] Brezin, E.; Hikami, S., Level spacing of random matrices in an external source, Phys. rev. E (3), 58, 6, 7176-7185, (1998), part A
[14] Brezin, E.; Hikami, S., Universal singularity at the closure of a gap in a random matrix theory, Phys. rev. E (3), 57, 4, 4140-4149, (1998)
[15] Daley, D.; Vere-Jones, D., An introduction to the theory of point processes: vol. I. elementary theory and methods, (2003), Springer-Verlag New York · Zbl 1026.60061
[16] Forrester, P.; Nordenstam, E., The anti-symmetric GUE minor process · Zbl 1191.15032
[17] Forrester, P.; Nagao, T., Determinantal correlations for classical projection processes · Zbl 0917.15018
[18] Johansson, K., Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of math. (2), 153, 1, 259-296, (2001) · Zbl 0984.15020
[19] Johansson, K., Discrete polynuclear growth and determinantal processes, Comm. math. phys., 242, 277-329, (2003) · Zbl 1031.60084
[20] Kerov, S.V., Distribution of symmetry types of high rank tensors, Zap. nauchn. sem. LOMI, J. soviet math. (New York), 41, 2, 995-999, (1988), (in Russian); English translation in · Zbl 0639.15008
[21] Kerov, S.V., Asymptotic representation theory of the symmetric group and its applications in analysis, Transl. math. monogr., vol. 219, (2003), Amer. Math. Soc. · Zbl 1031.20007
[22] Logan, B.F.; Shepp, L.A., A variational problem for random Young tableaux, Adv. math., 26, 206-222, (1977) · Zbl 0363.62068
[23] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), The Clarendon Press, Oxford University Press New York · Zbl 0487.20007
[24] Okounkov, A., Random matrices and random permutations, Int. math. res. not., 2000, 20, 1043-1095, (2000) · Zbl 1018.15020
[25] Okounkov, A., Symmetric functions and random partitions, (), 223-252 · Zbl 1017.05103
[26] Okounkov, A.; Olshanski, G., Asymptotics of Jack polynomials as the number of variables goes to infinity, Int. math. res. not., 1998, 13, 641-682, (1998) · Zbl 0913.33004
[27] Okounkov, A.; Reshetikhin, N., Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. amer. math. soc., 16, 3, 581-603, (2003) · Zbl 1009.05134
[28] Okounkov, A.; Reshetikhin, N., Random skew plane partitions and the pearcey process, Comm. math. phys., 269, 3, 571-609, (2007) · Zbl 1115.60011
[29] Olshanski, G., The problem of harmonic analysis on the infinite dimensional unitary group, J. funct. anal., 205, 2, 464-524, (2003) · Zbl 1036.43002
[30] Prähofer, M.; Spohn, H., Scale invariance of the PNG droplet and the Airy process, J. stat. phys., 108, 5-6, 1071-1106, (2002) · Zbl 1025.82010
[31] Tracy, C.; Widom, H., The pearcey process, Comm. math. phys., 263, 381-400, (2006) · Zbl 1129.82031
[32] Vershik, A.; Kerov, S., Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Soviet math. dokl., 18, 527-531, (1977) · Zbl 0406.05008
[33] Vershik, A.; Kerov, S., Characters and factor representations of the infinite unitary group, Soviet math. dokl., 26, 570-574, (1982) · Zbl 0524.22017
[34] Vershik, A.; Kerov, S., Asymptotics of the maximal and typical dimension of irreducible representations of symmetric group, Funct. anal. appl., 19, 1, 25-36, (1985)
[35] Voiculescu, D., Représentations factorielles de type \(\mathit{II}_1\) de \(U(\infty)\), J. math. pures appl., 55, 1-20, (1976) · Zbl 0352.22014
[36] Zhelobenko, D.P., Compact Lie groups and their representations, Transl. math. monogr., vol. 40, (1973), Amer. Math. Soc. Providence, RI, (in Russian); English translation in · Zbl 0228.22013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.