Let $(p_x, r_x)_{x \in \mathbb{Z}_+}$ be a sequence of independent and identically distributed random vectors such that $p_x > 0$, $r_x \geq 0$, and $p_x + r_x \leq 1$. Define the sequence of $2 \times 2$ random matrices:

$$A_x = \begin{pmatrix} \frac{1 - p_x}{p_x} & \frac{r_x}{p_x} \\ 1 & 0 \end{pmatrix}$$

By the Furstenberg-Kesten theorem, the top Lyapunov exponent $\lambda$ of this matrix sequence is defined almost surely as:

$$\lambda = \lim_{n \to \infty} \frac{1}{n} \log \| A_n A_{n-1} \cdots A_1 \|$$

\textbf{Open Problem:} Determine a closed-form algebraic expression for $\lambda$, or explicitly characterize the critical phase boundary where $\lambda = 0$, strictly in terms of the marginal joint probability distribution of $(p_1, r_1)$.

\medskip

This matrix product naturally arises when evaluating hitting times and recurrence criteria for 1D Random Walks in Random Environments (RWRE) with jumps of size 2.

\begin{thebibliography}{9}

\bibitem{bremont2002}
Br\'emont, J. (2002).
\newblock On some random walks on Z in random medium.
\newblock \emph{The Annals of Probability}, 30(3), 1266--1312.

\bibitem{solomon1975}
Solomon, F. (1975).
\newblock Random walks in a random environment.
\newblock \emph{The Annals of Probability}, 3(1), 1--31.

\end{thebibliography}