## The Annals of Probability

### Eigenvalue distributions of random permutation matrices

Kelly Wieand

#### Abstract

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 $(X -E[X])/( \Var (X))^ 1/2$ is asymptotically normally distributed. In addition, we show that for several fixed arcs $I_1,\ldots,I_m$, the corresponding random variables are jointly normal in the large $n$ limit.

#### Article information

Source
Ann. Probab. Volume 28, Number 4 (2000), 1563-1587.

Dates
First available in Project Euclid: 18 April 2002

http://projecteuclid.org/euclid.aop/1019160498

Digital Object Identifier
doi:10.1214/aop/1019160498

Mathematical Reviews number (MathSciNet)
MR1813834

Zentralblatt MATH identifier
01905954

Keywords
Permutations random matrices

#### Citation

Wieand, Kelly. Eigenvalue distributions of random permutation matrices. Ann. Probab. 28 (2000), no. 4, 1563--1587. doi:10.1214/aop/1019160498. http://projecteuclid.org/euclid.aop/1019160498.

#### References

• [1] Arratia, R., Barbour, A. D. and Tavar´e, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519-535.
• [2] Arratia, R. and Tavar´e, S. (1992). The cycle structure of random permutations. Ann. Probab. 20 1567-1591.
• [3] Arratia, R. and Tavar´e, S. (1992). Limit theorems for combinatorial structures via discrete process approximations. RandomStructures Algorithms 3 321-345.
• [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.
• [5] Barbour, A. D. and Tavar´e, S. (1994). A rate for the Erd¨os-Tur´an law. Combin. Probab. Comput. 3 167-176.
• [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.
• [8] Durrett, R. (1991). Probability: Theory and Examples. Brooks/Cole, Belmont, CA.
• [9] Feller, W. (1945). The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51 800-832.
• [10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
• [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. To appear.
• [13] Kuipers, L. and Niederreiter, H. (1974). UniformDistribution of Sequences. Wiley, New York.
• [14] P ´olya, G. and Szeg ¨o, G. (1972). Problems and Theorems in Analysis 1. Springer, Berlin.
• [15] Rains, E. (1997). High powers of random elements of compact lie groups. Probab. Theory Related Fields 107 219-241.
• [16] Riordan, J. (1958). An Introduction to Combinatorial Analysis. Wiley, New York.
• [17] Shepp, L. A. and Lloyd, S. P. (1966). Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357.
• [18] Soshnikov, A. (1998). Level spacings distribution for large random matrices: Gaussian fluctuations. Ann. Math. 148 573-617.
• [19] Wieand, K. (1998). Eigenvalue distributions of random matrices in the permutation group and compact lie groups. PhD dissertation, Harvard Univ.