×

Eigenvalue distributions of random permutation matrices. (English) Zbl 1044.15017

Summary: Let \(M\) be a randomly chosen \(n\times n\) permutation matrix. For a fixed arc of the unit circle, let \(X\) be the number of eigenvalues of \(M\) which lie in the specified arc. We calculate the large \(n\) asymptotics for the mean and variance of \(X\), and show that \((X-E[X])/(\text{Var} (X))^{1/2}\) is asymptotically normally distributed. In addition, we show that for several fixed arcs \(I_1,\dots,I_m\), the corresponding random variables are jointly normal in the large \(n\) limit.

MSC:

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519-535. · Zbl 0756.60006
[2] Arratia, R. and Tavaré, S. (1992). The cycle structure of random permutations. Ann. Probab. 20 1567-1591. · Zbl 0759.60007
[3] Arratia, R. and Tavaré, S. (1992). Limit theorems for combinatorial structures via discrete process approximations. RandomStructures Algorithms 3 321-345. · Zbl 0758.60009
[4] Baik, J., Deift, P. and Johansson, K. (1998). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178. JSTOR: · Zbl 0932.05001
[5] Barbour, A. D. and Tavaré, S. (1994). A rate for the Erdös-Turán law. Combin. Probab. Comput. 3 167-176. · Zbl 0805.60008
[6] Costin, O. and Lebowitz, J. (1995). Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75 69-72.
[7] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31 49-61. JSTOR: · Zbl 0807.15015
[8] Durrett, R. (1991). Probability: Theory and Examples. Brooks/Cole, Belmont, CA. · Zbl 0709.60002
[9] Feller, W. (1945). The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51 800-832. · Zbl 0060.28702
[10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0219.60003
[11] Goncharov, V. (1962). Du domaine d’analyse combinatoire. Amer. Math. Soc. Transl. Ser. 2 19 1-46.
[12] Hambly, B., Keevash, P., O’Connell, N. and Stark, D. (2000). The characteristic polynomial of a random permutation matrix. Stochastic Process. Appl. · Zbl 1047.60013
[13] Kuipers, L. and Niederreiter, H. (1974). UniformDistribution of Sequences. Wiley, New York. · Zbl 0281.10001
[14] P ólya, G. and Szeg ö, G. (1972). Problems and Theorems in Analysis 1. Springer, Berlin. · Zbl 0236.00003
[15] Rains, E. (1997). High powers of random elements of compact lie groups. Probab. Theory Related Fields 107 219-241. · Zbl 0868.60012
[16] Riordan, J. (1958). An Introduction to Combinatorial Analysis. Wiley, New York. · Zbl 0078.00805
[17] Shepp, L. A. and Lloyd, S. P. (1966). Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357. JSTOR: · Zbl 0156.18705
[18] Soshnikov, A. (1998). Level spacings distribution for large random matrices: Gaussian fluctuations. Ann. Math. 148 573-617. JSTOR: · Zbl 0944.60060
[19] Wieand, K. (1998). Eigenvalue distributions of random matrices in the permutation group and compact lie groups. PhD dissertation, Harvard Univ.
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.