We study harmonic chains with i.i.d. random spring constants $K_n$ and i.i.d. random masses $m_n$. We introduce a combinatorial approach which allows us to derive a compact and general approximate formula for the complex Lyapunov exponent, in terms of the solutions of two transcendental equations involving the distributions of the spring constants and the masses. Our result makes easy the asymptotic analysis of the low-frequency properties of the eigenmodes (spectral density and localization) for arbitrary disorder distribution, as well as their high-frequency properties. We apply the method to the case of power-law distributions $p\left(K\right)=\mu K^{-1+\mu }$ with $0<K<1$ and $q\left(m\right)=\nu m^{-1-\nu }$ with $m>1$ (with $\mu ,\nu >0$). At low frequency, the spectral density presents the power law $\varrho (\omega \to 0)\sim {\omega }^{2\eta -1}$, where the exponent $\eta$ exhibits first-order phase transitions on the line $\mu =1$ and on the line $\nu =1$. The exponent of the nondisordered chain ($\eta =1/2$) is recovered when $\langle K_n^{-1}\rangle$ and $\langle m_n\rangle$ are both finite, i.e., $\mu >1$ and $\nu >1$. The Lyapunov exponent (inverse localization length) shows also a power-law behavior $\gamma ({\omega }^2\to 0)\sim {\omega }^{2\zeta }$, where the exponent $\zeta$ exhibits several phase transitions: the exponent is $\zeta =\eta$ for $\mu <1$ or $\nu <1$ ($\langle K_n^{-1}\rangle$ or $\langle m_n\rangle$ infinite) and $\zeta =1$ when $\mu >2$ and $\nu >2$ ($\langle K_n^{-2}\rangle$ and $\langle m_n^2\rangle$ both finite). In the intermediate region it is given by $\zeta =\min (\mu ,\nu )/2$. On the transition lines, $\varrho \left(\omega \right)$ and $\gamma \left({\omega }^2\right)$ receive logarithmic corrections. Finally, we illustrate the versatility of our combinatorial approach by considering the case of the Anderson model with random couplings (a model known to be mapped onto the random spring chain for “Dyson type I” disorder).

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• Anderson localization
• 1-dimensional systems
• Disordered systems
• Random & disordered media
• Random matrix theory