{"id":1647,"date":"2016-10-28T08:31:35","date_gmt":"2016-10-28T08:31:35","guid":{"rendered":"http:\/\/lptms.u-psud.fr\/christophe_texier\/?page_id=1647"},"modified":"2020-10-06T08:57:01","modified_gmt":"2020-10-06T08:57:01","slug":"products-of-random-matrices","status":"publish","type":"page","link":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/random-matrix-theory\/products-of-random-matrices\/","title":{"rendered":"Products of random matrices"},"content":{"rendered":"<p>The study of a wave equation in an inhomogeneous medium, such as an electronic wave in a metallic wave guide, may be formulated in terms of <strong>transfer matrices<\/strong>. Within this frame, the analysis of 1D or quasi-1D (multichannel) random media leads to consider random matrix products:<\/p>\n<p><a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/files\/2016\/10\/disordered-wire.png\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1861 aligncenter\" src=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/files\/2016\/10\/disordered-wire-300x116.png\" alt=\"disordered-wire\" width=\"419\" height=\"162\" srcset=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/files\/2016\/10\/disordered-wire-300x116.png 300w, http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/files\/2016\/10\/disordered-wire.png 452w\" sizes=\"auto, (max-width: 419px) 100vw, 419px\" \/><\/a><\/p>\n<p>(see page on <a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/recherche\/some-properties-of-the-one-dimensional-schrodinger-operator-with-random-potential\/\">1D disordered QM<\/a>). In particular, in 2010, we have found the connection between general matrix products in SL(2,R) and a model of generalised point scatterers:<\/p>\n<ul>\n<li>Alain Comtet, Christophe Texier and Yves Tourigny,<br \/>\n<strong>Products of random matrices and generalised quantum point scatterers<\/strong><br \/>\n<a href=\"http:\/\/link.springer.com\/article\/10.1007%2Fs10955-010-0005-x\">J. Stat. Phys. <strong>140<\/strong>, 427-466 (2010)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1004.2415\">cond-mat arXiv:1004.2415.<\/a><\/li>\n<li>Alain Comtet,\u00a0 Christophe Texier and Yves Tourigny,<br \/>\n<strong>Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices<\/strong>, [Review article]<br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1751-8121\/46\/25\/254003\/\">J. Phys. A : Math. Gen. <strong>46<\/strong>, 254003 (2013).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1207.0725\">cond-mat arXiv:1207.0725<\/a><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>General products of random 2*2 matrices in the continuum limit (random matrices close to the identity)<\/h3>\n<p>We find analytical solutions for the Lyapunov exponent (i.e. the characteristic function) for most general products of random matrices of SL(2,R), in the continuum limit (matrices close to the identity).<\/p>\n<p>Many of one-dimensional disordered problems may be mapped onto random matrix products. E.g. : Ising chain with random couplings and random magnetic field, in this case the Lyapunov exponent is interpreted as the free energy per site ; 1D quantum mechanics with disorder for which the Lyapunov exponent provides a measure of the localisation (precisely, the Lyapunov exponent is the inverse localisation length \u03b3=1\/\u03be).<\/p>\n<p>Our general formulation allows us to exhaust all possible 1D disordered models mapped onto random matrix products of SL(2,R) ; we have provided a classification of solutions (we recover known result and find several new solvable cases) that can be understand as a <strong>classification of 1D continuum disordered models<\/strong>.<\/p>\n<ul>\n<li>Alain Comtet, Jean-Marc Luck, Christophe Texier and Yves Tourigny,<br \/>\n<strong> The Lyapunov exponent of products of random 2*2 matrices close to the identity,<\/strong><br \/>\n<a href=\"http:\/\/link.springer.com\/article\/10.1007%2Fs10955-012-0674-8\">J. Stat. Phys. <strong>150<\/strong>, 13-65 (2013).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1208.6430\">math-ph arXiv:1208.6430<\/a><\/li>\n<\/ul>\n<h3>Fluctuations of random matrix products of SL(2,R) : Generalised Lyapunov exponent<\/h3>\n<p>Besides the study of the Lyapunov exponent, which characterises the average of the logarithm of a product of random matrices, the fluctuations may also be of interest: several motivations from physics are: the question of Single Parameter Scaling, the analysis of conductance fluctuations, etc. The study of fluctuations requires to study the <em>generalised Lyapunov exponent<\/em> or its expansion (cumulants):<\/p>\n<p style=\"padding-left: 90px\">\u039b(<em>q<\/em>) = lim<sub><em>N<\/em>\u2192\u221e<\/sub>(1\/<em>N<\/em>) ln&lt;||\u03a0<em><sub>N<\/sub><\/em>||<em><sup>q<\/sup><\/em>&gt;<\/p>\n<p style=\"padding-left: 180px\"><a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/files\/2013\/03\/armp4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1377\" src=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/files\/2013\/03\/armp4-300x189.png\" alt=\"armp4\" width=\"300\" height=\"189\" srcset=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/files\/2013\/03\/armp4-300x189.png 300w, http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/files\/2013\/03\/armp4.png 545w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<ul>\n<li>Kabir Ramola and Christophe Texier,<br \/>\n<strong>Fluctuations of random matrix products and 1D Dirac equation with random mass<\/strong>,<br \/>\n<a href=\"http:\/\/link.springer.com\/article\/10.1007%2Fs10955-014-1082-z\"> J. Stat. Phys. <strong>157<\/strong>, 497-514 (2014)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1402.6943\">cond-mat arXiv:1402.6943<\/a><br \/>\nSome integral formulae for the variance are obtained, for few specific cases of random 2*2 matrices.<\/li>\n<li>Christophe Texier,<br \/>\n<strong>Fluctuations of the product of random matrices and generalized Lyapunov exponent<\/strong>,<br \/>\n<a href=\"https:\/\/link.springer.com\/article\/10.1007%2Fs10955-020-02617-w\">J. Stat. Phys (2020)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1907.08512\">cond-mat arXiv:1907.08512<\/a><br \/>\nA general formalism is developed for any random matrix product of SL(2,R)<\/li>\n<li>Christophe Texier,<br \/>\n<strong>Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation<\/strong>,<br \/>\n<a href=\"https:\/\/iopscience.iop.org\/article\/10.1209\/0295-5075\/131\/17002\">Europhys. Lett. <strong>131<\/strong>, 17002 (2020)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1910.01989\">cond-mat arXiv:1910.01989<\/a><br \/>\nUsing the general formalism of the previous article, the Single Parameter Scaling for Anderson localization is proven within a general framework, and extended to potentials with large fluctuations (such that &lt;V<sup>2<\/sup>&gt;=\u221e)<\/li>\n<li>Alain Comtet, Christophe Texier and Yves Tourigny,<br \/>\n<strong>Representation theory and products of random matrices in SL(2,R)<\/strong>,<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1911.00117\">math-ph arXiv:1911.00117<\/a><br \/>\nIt is shown how concepts of representation theory can be used for the study of random matrix products and disordered systems.<\/li>\n<\/ul>\n<h3>The multichannel (quasi 1D) case : random matrix products in the chiral symmetry classes and the Dirac equation with a random matrix mass<\/h3>\n<ul>\n<li>Aur\u00e9lien Grabsch and Christophe Texier,<br \/>\n<strong>Topological phase transitions\u00a0 in the 1D multichannel Dirac equation with random mass and a random matrix model<\/strong>,<br \/>\n<a href=\"http:\/\/iopscience.iop.org\/article\/10.1209\/0295-5075\/116\/17004\/meta;jsessionid=2307AFF1973D4ACBF1E032CB85A50FD8.c4.iopscience.cld.iop.org\">Europhys. Lett. <strong>116<\/strong>, 17004 (2016)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1506.05322\"> cond-mat arXiv:1506.05322<\/a><\/li>\n<\/ul>\n<p>We have studied the Dirac equation for <em>N<\/em> channels with a random (matricial mass). This corresponds to studying certain random matrix products of the symplectic group\u00a0 Sp(N,R) in the continuum limits.<\/p>\n<p>cf. page \u201c <a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/recherche\/some-properties-of-the-one-dimensional-schrodinger-operator-with-random-potential\/\">1D-disordered systems<\/a> \u201d for details<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The study of a wave equation in an inhomogeneous medium, such as an electronic wave in a metallic wave guide, may be formulated in terms of transfer matrices. Within this frame, the analysis of 1D or quasi-1D (multichannel) random media &hellip; <a href=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/random-matrix-theory\/products-of-random-matrices\/\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"parent":1028,"menu_order":22,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1647","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/1647","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/comments?post=1647"}],"version-history":[{"count":21,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/1647\/revisions"}],"predecessor-version":[{"id":2461,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/1647\/revisions\/2461"}],"up":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/1028"}],"wp:attachment":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/media?parent=1647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}