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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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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 Wolowski
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: l.wolowski@bristol.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04316-X
PII: S 0002-9947(08)04316-X
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