Consider the following self-adjoint matrix

$A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some interval $[-\varepsilon,\varepsilon].$

Now, take the product

$$M_n = A_{X_n} \cdot...\cdot A_{X_1}$$

where $X_1,...,X_n$ are iid copies of $X.$

We observe that $\operatorname{det}(M_n)=(-1)^n.$ Thus, $M_n$ has an eigenvalue that is at least of modulus $1.$

We then consider the Lyapunov exponent $$ \mu = \lim_{n \to \infty}\frac{1}{n} \log \Vert M_n\Vert.$$

By the above reasoning we have that $\mu \ge 0.$

Is it true that $\mu>0?$