Random Fibonacci sequences and the number $1.13198824\dots$

Abstract:

For the familiar Fibonacci sequence (defined by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n>2$), $f_n$ increases exponentially with $n$ at a rate given by the golden ratio $(1+\sqrt {5})/2=1.61803398\ldots$. But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by $t_1=t_2=1$, and for $n>2$, $t_n = \pm t_{n-1} \pm t_{n-2}$, where each $\pm$ sign is independent and either $+$ or $-$ with probability $1/2$ — it is not even obvious if $\vert {t_n}\vert$ should increase with $n$. Our main result is that \begin{equation*} \sqrt [n]{\vert {t_n}\vert } \rightarrow 1.13198824\ldots \:\:\: \text {as}\:\:\: n \rightarrow \infty \end{equation*} with probability $1$. Finding the number $1.13198824\ldots$ involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.