Fluctuations of random matrix products and 1D Dirac equation with random mass

Abstract

We study the fluctuations of certain random matrix products $Π_N=M_N\cdots M_2M_1$ of $\mathrm{SL}(2,\mathbb{R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices $M_n$'s are close to the identity matrix, we obtain convenient integral representations for the variance $Γ_2=\lim_{N\to\infty}\mathrm{Var}(\ln||Π_N||)/N$. The case studied exhibits a saturation of the variance at low energy $\varepsilon$ along with a vanishing Lyapunov exponent $Γ_1=\lim_{N\to\infty}\ln||Π_N||/N$, leading to the behaviour $Γ_2/Γ_1\sim\ln(1/|\varepsilon|)\to\infty$ as $\varepsilon\to0$. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.