Fluctuations of Random Matrix Products and 1D Dirac Equation with Random Mass

Abstract

We study the fluctuations of certain random matrix products, 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 →∞ Var(ln ||Π N ||)/N . The case studied exhibits a saturation of the variance at low energy ε along with a vanishing Lyapunov exponent Γ 1 = lim N →∞ ln ||Π N ||/N , leading to the behaviour Γ 2 /Γ 1 ∼ ln(1/|ε|) → ∞ as ε → 0. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

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Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model

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