Weak Disorder in Fibonacci Sequences
E. Ben-Naim and P.L. Krapivsky
We study how weak disorder affects the growth of the Fibonacci
series. We introduce a family of stochastic sequences that grow by
the normal Fibonacci recursion with probability $1-\epsilon$, but
follow a different recursion rule with a small probability
$\epsilon$. We focus on the weak disorder limit and obtain the
Lyapunov exponent, that characterizes the typical growth of the
sequence elements, using perturbation theory. The limiting
distribution for the ratio of consecutive sequence elements is
obtained as well. A number of variations to the basic Fibonacci
recursion including shift, doubling, and copying are considered.
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