Abstract.
The distribution of the shape λ of the semi-standard tableau of a random word in k letters is asymptotically given by the distribution of the spectrum of a random traceless k×k Gaussian Unitary Ensemble (GUE) matrix provided that these letters are independent with uniform distribution. Kuperberg (2002) conjectured that this result by Johansson (2001) remains valid if the letters of the word are generated by an irreducible Markov chain on the alphabet with cyclic transition matrix. In this paper we give a proof of this conjecture for an alphabet with k=2 letters.
References
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, 1119–1178 (1999)
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 (2000a)
Baik, J., Deift, P., Johansson, K.: Addendum to: ‘‘On the distribution of the length of the second row of a Young diagram under Plancherel measure’’. Geom. Funct. Anal. 10 (6), 1606–1607 (2000b)
Borodin, A., Okounkov, A., and Olshanskii, G.: Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (3), 481–515 (2000)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, Wiley, New York, London, Sydney 1968, pp. 509
Fulton, W.: Young tableaux: with applications to representation theory and geometry, Cambridge University Press, Cambridge, 1997, pp. 260
Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (301), 13–30 (1963)
Its, A., Tracy, G., Widom, H.: Toeplitz determinants, and integrable systems. I. Math. Sci. Res. Inst. Publ. 40, 245–258 (2001)
Its, A., Tracy, G., Widom, H.: Toeplitz determinants, and integrable systems. II. Phys. D 152/153, 199–224 (2001)
Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 (1), 259–296 (2001)
Kuperberg, G.: Random words, quantum statistics, central limits, random matrices. Methods Appl. Anal. 9 (1), 99–118 (2002)
Löwe, M., Merkl, F.: Moderate deviations for longest increasing subsequences: the upper tail. Comm. Pure Appl. Math. 54 (12), 1488–1520 (2001)
Löwe, M., Merkl, F., and Rolles, S.: Moderate deviations for longest increasing subsequences: the lower tail. J. Theor. Probab. 15 (4), 1031–1047 (2002)
Mumford, D.: Tata Lectures on Theta I Birkhäuser, Boston–Basel–Stuttgart, 1983
O’Connelli, N.: Conditioned random walks and the RSK correspondence. J. Phys. A: Math. Gen. 36, 3049–3066 (2003)
Okounkov, A.: Random matrices and random permutations. Internat. Math. Res. Notices (20), 1043–1095 (2000)
Stanley, R.P.: Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999, pp. 581
Tracy, C.A., Widom, H.: On the distributions of the lenths of the longest monotone subsequences in random words. Probab. Theory Related Fields 119 (3), 350–380 (2001)
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Research supported by DFG GO-420/3-3 in Bielefeld.
Research supported by INTAS 99-00317, RFBR–DFG 99-01-04027.
Mathematics Subject Classification (2000): 82B41, 60C05, 60F05, 60F10
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Chistyakov, G., Götze, F. Distribution of the shape of Markovian random words. Probab. Theory Relat. Fields 129, 18–36 (2004). https://doi.org/10.1007/s00440-003-0327-6
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DOI: https://doi.org/10.1007/s00440-003-0327-6
Key words or phrases
- Random words
- central limits
- random matrices
- Markov’s chain