Lyapounov exponents of products of random matrices : weak disorder expansion. - Application to localisation

B. Derrida , K. Mecheri , J.L. Pichard

We derive the weak disorder expansion of the Lyapounov exponents of a product of random matrices. The condition for this expansion to be valid is that in the limit of zero disorder, the matrix has all its eigenvalues with different moduli. As an example we study the problem of localisation on strips in the limit of weak disorder. We show that our expansion agrees very well with numerical simulations in the region where the condition on the moduli is satisfied which corresponds to energies outside the conduction band. In that region, we find a limiting density of Lyapounov exponents when the strip width goes to infinity. Inside the band, our expansion cannot be valid unless one adds an imaginary part to the energy.

Mots clés: lattice theory and statistics; Lyapunov methods; random processes

Domaines: Articles anciens

B. Derrida, K. Mecheri, J.L. Pichard. Lyapounov exponents of products of random matrices : weak disorder expansion. - Application to localisation. Journal de Physique, 1987, 48 (5), pp.733-740. ⟨10.1051/jphys:01987004805073300⟩. ⟨jpa-00210492⟩