Explicit invariant measures for products of random matrices
HTML articles powered by AMS MathViewer
- by Jens Marklof, Yves Tourigny and Lech Wołowski PDF
- Trans. Amer. Math. Soc. 360 (2008), 3391-3427 Request permission
Abstract:
We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of $\text {SL}(2,{\mathbb C})$. 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.References
- M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, 1964
- P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492-1505 (1958).
- George A. Baker Jr. and Peter Graves-Morris, Padé approximants, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 59, Cambridge University Press, Cambridge, 1996. MR 1383091, DOI 10.1017/CBO9780511530074
- Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 538168
- Évelyne Bernadac, Random continued fractions and inverse Gaussian distribution on a symmetric cone, J. Theoret. Probab. 8 (1995), no. 2, 221–259. MR 1325851, DOI 10.1007/BF02212879
- Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
- René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
- René Carmona, Abel Klein, and Fabio Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41–66. MR 872140
- Jean-François Chamayou and Gérard Letac, Explicit stationary distributions for compositions of random functions and products of random matrices, J. Theoret. Probab. 4 (1991), no. 1, 3–36. MR 1088391, DOI 10.1007/BF01046992
- A. Comtet, Private communication.
- Persi Diaconis and David Freedman, Iterated random functions, SIAM Rev. 41 (1999), no. 1, 45–76. MR 1669737, DOI 10.1137/S0036144598338446
- Freeman J. Dyson, The dynamics of a disordered linear chain, Phys. Rev. (2) 92 (1953), 1331–1338. MR 59210
- Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428. MR 163345, DOI 10.1090/S0002-9947-1963-0163345-0
- H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469. MR 121828, DOI 10.1214/aoms/1177705909
- I.S. Gradshteyn and I.M. Ryzhik, Table of integrals series and products, Academic Press, New York, 1965.
- D.C. Herbert and R. Jones, Localized states in disordered systems, J. Phys. C. Solid State Phys. 4, (1971) 1145-1161.
- Hervé Kunz and Bernard Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), no. 2, 201–246 (French, with English summary). MR 597748
- Gérard Letac and Vanamamalai Seshadri, A characterization of the generalized inverse Gaussian distribution by continued fractions, Z. Wahrsch. Verw. Gebiete 62 (1983), no. 4, 485–489. MR 690573, DOI 10.1007/BF00534200
- Gérard Letac and Vanamamalai Seshadri, A random continued fraction in $\textbf {R}^{d+1}$ with an inverse Gaussian distribution, Bernoulli 1 (1995), no. 4, 381–393. MR 1369168, DOI 10.2307/3318490
- 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.
- A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405, DOI 10.1007/978-1-4757-2539-1
- 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.
- Y. Tourigny and P. G. Drazin, The dynamics of Padé approximation, Nonlinearity 15 (2002), no. 3, 787–805. MR 1901106, DOI 10.1088/0951-7715/15/3/316
- G. N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.
Additional Information
- Jens Marklof
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- Email: j.marklof@bristol.ac.uk
- Yves Tourigny
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- Email: y.tourigny@bristol.ac.uk
- Lech Wołowski
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- Email: l.wolowski@bristol.ac.uk
- 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)
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 3391-3427
- MSC (2000): Primary 15A52, 11J70
- DOI: https://doi.org/10.1090/S0002-9947-08-04316-X
- MathSciNet review: 2386231