zbMATH — the first resource for mathematics

Elementary divisors and determinants of random matrices over a local field. (English) Zbl 1075.15500
Summary: We consider the elementary divisors and determinant of a uniformly distributed \(n \times n\) random matrix with entries in the ring of integers of an arbitrary local field. We show that the sequence of elementary divisors is in a simple bijective correspondence with a Markov chain on the non-negative integers. The transition dynamics of this chain do not depend on the size of the matrix. As \(n \to \infty \), all but finitely many of the elementary divisors are 1, and the remainder arise from a Markov chain with these same transition dynamics. We also obtain the distribution of the determinant of \(M_n\) and find the limit of this distribution as \(n \to \infty \). Our formulae have connections with classical identities for \(q\)-series, and the \(q\)-binomial theorem, in particular.

15B52 Random matrices (algebraic aspects)
60B99 Probability theory on algebraic and topological structures
12J25 Non-Archimedean valued fields
05A30 \(q\)-calculus and related topics
Full Text: DOI
[1] Abdel-Ghaffar, K.A.S., The determinant of random power series matrices over finite fields, Linear algebra appl., 315, 1-3, 139-144, (2000) · Zbl 0964.15016
[2] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge
[3] Balakin, G.V., The distribution of the rank of random matrices over a finite field, Teor. verojatnost. i primenen., 13, 631-641, (1968) · Zbl 0184.41602
[4] Blömer, J.; Karp, R.; Welzl, E., The rank of sparse random matrices over finite fields, Random struct. algorithms, 10, 4, 407-419, (1997) · Zbl 0877.15027
[5] Brennan, J.P.; Wolfskill, J., Remarks: “on the probability that the determinant of an n×n matrix over a finite field vanishes” [discrete math. 51(3) (1984) 311-315; MR 86a:15012] by A. mukhopadhyay, Discrete math., 67, 3, 311-313, (1987)
[6] Brent, R.P.; McKay, B.D., Determinants and ranks of random matrices over Zm, Discrete math., 66, 1-2, 35-49, (1987) · Zbl 0628.15010
[7] Cooper, C., 2000a. On the distribution of rank of a random matrix over a finite field. In: Proceedings of the Ninth International Conference “Random Structures and Algorithms”, Vol. 17, Poznan, 1999, pp. 197-212.
[8] Cooper, C., On the rank of random matrices, Random struct. algorithms, 16, 2, 209-232, (2000) · Zbl 0953.15025
[9] Evans, S.N., 2001. Local fields, Gaussian measures, and Brownian motions. In: Taylor, J.C. (Ed.), Topics in Probability and Lie Groups: Boundary Theory, American Mathematical Society, Providence, RI, pp. 11-50. · Zbl 0989.60039
[10] Fulman, J., Cycle indices for the finite classical groups, J. group theory, 2, 3, 251-289, (1999) · Zbl 0943.20048
[11] Fulman, J., Random matrix theory over finite fields, Bull. amer. math. soc. (N.S.), 39, 1, 51-85, (2002), (electronic) · Zbl 0984.60017
[12] Hansen, J.C.; Schmutz, E., How random is the characteristic polynomial of a random matrix?, Math. proc. Cambridge philos. soc., 114, 3, 507-515, (1993) · Zbl 0793.15013
[13] Jacobson, N., Basic algebra. I, (1985), Freeman New York · Zbl 0557.16001
[14] Kozlov, M.V., On the rank of matrices with random Boolean elements, Soviet math. dokl., 7, 1048-1051, (1966) · Zbl 0171.38702
[15] Kung, J.P.S., The cycle structure of a linear transformation over a finite field, Linear algebra appl., 36, 141-155, (1981) · Zbl 0477.05008
[16] Levit.skaya, A.A., 1991. Limit distribution of the rank of a random Boolean matrix with a fixed number of units in the rows. Dokl. Akad. Nauk Ukrain. SSR (8), 49-52, 183.
[17] Mukhopadhyay, A., On the probability that the determinant of an n×n matrix over a finite field vanishes, Discrete math., 51, 3, 311-315, (1984) · Zbl 0553.15003
[18] Pólya, G., On the number of certain lattice polygons, J. combin. theory, 6, 102-105, (1969) · Zbl 0327.05010
[19] Rawlings, D., Absorption processesmodels for q-identities, Adv. appl. math., 18, 2, 133-148, (1997) · Zbl 0867.05003
[20] Rawlings, D., A probabilistic approach to some of Euler’s number-theoretic identities, Trans. amer. math. soc., 350, 7, 2939-2951, (1998) · Zbl 0902.60092
[21] Schikhof, W.H., Ultrametric calculus, (1984), Cambridge University Press Cambridge, (An introduction to p-adic analysis) · Zbl 0553.26006
[22] Schützenberger, M.P., Une interprétation de certaines solutions de l’équation fonctionnellef(x+y)=F(x)F(y), C. R. acad. sci. Paris, 236, 352-353, (1953) · Zbl 0051.09401
[23] Stong, R., Some asymptotic results on finite vector spaces, Adv. appl. math., 9, 2, 167-199, (1988) · Zbl 0681.05004
[24] Taibleson, M.H., Fourier analysis on local fields, (1975), Princeton University Press Princeton, NJ · Zbl 0319.42011
[25] Waterhouse, W.C., How often do determinants over finite fields vanish?, Discrete math., 65, 1, 103-104, (1987) · Zbl 0623.15001
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.