Explicit invariant measures for products of random matrices

Simple item page
dc.creatorMarklof, Jens
dc.creatorTourigny, Yves
dc.creatorWolowski, Lech
dc.date2005-07-28
dc.date2006-03-18
dc.date.accessioned2026-07-07T06:42:18Z
dc.date.available2026-07-07T06:42:18Z
dc.descriptionWe construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2,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 and 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 Schroedinger 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.
dc.descriptionLaTeX, 37 pages, 5 figures, section 6 and references updated
dc.identifierhttps://arxiv.org/abs/math-ph/0507069
dc.identifierhttp://arxiv.org/abs/math-ph/0507069
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/101989
dc.subjectMathematical Physics
dc.subjectProbability
dc.subjectSpectral Theory
dc.subject82B44; 15A52; 11J70
dc.titleExplicit invariant measures for products of random matrices
dc.typetext

Files

Collections

arXiv