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 leads to consider random matrix products:
(see page on 1D disordered QM). In particular, in 2010, we have found the connection between general matrix products in SL(2,R) and a model of generalised point scatterers:
- Alain Comtet, Christophe Texier and Yves Tourigny,
Products of random matrices and generalised quantum point scatterers
J. Stat. Phys. 140, 427-466 (2010)
General products of random 2*2 matrices in the continuum limit (random matrices close to the identity)
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).
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 γ=1/ξ).
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 classification of 1D continuum disordered models.
- Alain Comtet, Jean-Marc Luck, Christophe Texier and Yves Tourigny,
The Lyapunov exponent of products of random 2*2 matrices close to the identity,
J. Stat. Phys. 150, 13-65 (2013).
Fluctuations of certain random matrix products of SL(2,R)
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. This requires to study the generalised Lyapunov exponent or its expansion (cumulants).
- Kabir Ramola and Christophe Texier,
Fluctuations of random matrix products and 1D Dirac equation with random mass,
J. Stat. Phys. 157, 497-514 (2014)
The multichannel (quasi 1D) case : random matrix products in the chiral symmetry classes and the Dirac equation with a random matrix mass
- Aurélien Grabsch and Christophe Texier,
Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model,
Europhys. Lett. 116, 17004 (2016)
We have studied the Dirac equation for N channels with a random (matricial mass). We have obtained the phase diagram (below), which presents N+1 sectors in the half plane (mass,disorder) :
cf. page “ 1D-disordered systems ”