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A reasonable first estimate is $\ell$ (the size of $L$) is approximately $4\log n/|\log p|$. To get this, consider choosing a sequence of $k$ rows $i_1,\ldots,i_k$ and $k$ columns, $j_1,\ldots j_k$. …

The theory is right! If $H_i$ has negative Lyapunov exponents, then images of any vector under the transpose system will be exponentially close to a single vector (everything is being compressed in th …

Since you haven't given a distribution, let me make an observation giving the right form of the answer in the case where the $D_{xy}$ are independent uniform $[0,1]$ random variables. I want to clai …

So this is correct. The theorem that you need is the multiplicative ergodic theorem. Expressing it in your language, it states that $\frac 1n\log s_i(A_n)\to\lambda_i$, where $s_i$ is the $i$th singul …

If $A_1=\begin{pmatrix}1&0\\1&1\end{pmatrix}$ and $A_2=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, and the measure on $\{A_1,A_2\}^{\mathbb Z}$ is the $(\frac 12,\frac 12)$ Bernoulli measure, then there is …