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Explicit invariant measures for products of random matrices

Authors: Jens Marklof, Yves Tourigny and Lech Wolowski
Journal: Trans. Amer. Math. Soc. 360 (2008), 3391-3427
MSC (2000): Primary 15A52, 11J70
Published electronically: January 4, 2008
MathSciNet review: 2386231
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Abstract: We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL$ (2,{\mathbb{C}})$. The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrödinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

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  • 1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, 1964
  • 2. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492-1505 (1958).
  • 3. G. A. Baker and P. Graves-Morris, Padé Approximants, Cambridge University Press, Cambridge, 1996. MR 1383091 (97h:41001)
  • 4. C. Bender & S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. MR 538168 (80d:00030)
  • 5. E. Bernadac, Random continued fractions and inverse gaussian distribution on a symmetric cone, J. Th. Prob. 8 (1995) 221-259. MR 1325851 (96d:60006)
  • 6. P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, 1985. MR 886674 (88f:60013)
  • 7. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, 1990. MR 1102675 (92k:47143)
  • 8. R. Carmona, A. Klein and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41-66. MR 872140 (88f:82027)
  • 9. J. F. Chamayou and G. Letac, Explicit stationary distributions for compositions of random functions and products of random matrices, J. Th. Prob. 4 (1991) 3-36. MR 1088391 (92e:60014)
  • 10. A. Comtet, Private communication.
  • 11. P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev 41 (1999) 41-66. MR 1669737 (2000c:60102)
  • 12. F. J. Dyson, The dynamics of a disordered linear chain, Phys. Rev. 92 (1953). MR 0059210 (15:492c)
  • 13. H. Furstenberg, Non commuting random products, Trans. Amer. Math. Soc. 108 (1963) 377-428. MR 0163345 (29:648)
  • 14. H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960) 377-428. MR 0121828 (22:12558)
  • 15. I.S. Gradshteyn and I.M. Ryzhik, Table of integrals series and products, Academic Press, New York, 1965.
  • 16. D.C. Herbert and R. Jones, Localized states in disordered systems, J. Phys. C. Solid State Phys. 4, (1971) 1145-1161.
  • 17. H. Knuz, B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980), no. 2, 201-246. MR 597748 (83f:39003)
  • 18. G. Letac and V. Seshadri, A characterisation of the generalised inverse gaussian distribution by continued fractions, Z. Wahrsch. Verw. Gebiete 62 (1983) 485-489. MR 690573 (84f:62021)
  • 19. G. Letac and V. Seshadri, A random continued fraction in $ {\mathbb{R}}^{d+1}$ with an inverse Gaussian distribution, Bernoulli 1 (1995) 381-393. MR 1369168 (96m:60031)
  • 20. P. Lloyd, Exactly solvable model of electronic states in a three-dimensional disordered Hamiltonian: non-existence of localized states, J. Phys. C. Solid State Phys. 2 (1969) 1717-1725.
  • 21. A. N. Shiryaev, Probability (2nd ed.), Springer, New-York, 1996. MR 1368405 (97c:60003)
  • 22. D.J. Thouless, A relation between the density of states and range of localization for one dimensional random systems, J. Phys. C. Solid State Phys. 5 (1972) 78-81.
  • 23. Y. Tourigny and P. G. Drazin, The dynamics of Padé approximation, Nonlinearity 15, 787-805 (2002). MR 1901106 (2003g:41021)
  • 24. G. N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.

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Additional Information

Jens Marklof
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Yves Tourigny
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Lech Wolowski
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Keywords: Products of random matrices, continued fraction
Received by editor(s): August 9, 2005
Received by editor(s) in revised form: March 20, 2006
Published electronically: January 4, 2008
Additional Notes: The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council (United Kingdom) under Grant GR/S87461/01 and an Advanced Research Fellowship (JM)
Article copyright: © Copyright 2008 American Mathematical Society

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