REVIEW
Weak Disorder in Fibonacci Sequences
Statistical Mechanics · Submitted 6 Mar 2006 · 16 references
Pith's verdict: Clean, self-contained perturbation theory for weak disorder in a Fibonacci sequence; the first-order Lyapunov exponent and the exact doubling-model solution are correct, and the paper earns serious refereeing.
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Referee report
The paper
The paper claims
Rare errors shift Fibonacci growth at a rate set by the golden ratio.
The reading
What carries the argument
The ratio $r_n=x_n/x_{n-1}$ obeys the random map $r_{n+1}=1+1/r_n$ with probability $1-\epsilon$ and $r_{n+1}=1+1/(r_n r_{n-1})$ with probability $\epsilon$. In the disorder-free limit the map's stable fixed point is the golden ratio $\lambda$, and the paper's mechanism is to treat errors as rare perturbations that kick the ratio off $\lambda$, after which the normal map drags it back along a deterministic sequence $\rho_n$. The distribution in Eq. (13) is built from that sequence with geometric weights $\epsilon(1-\epsilon)^{n-1}$; summing $\ln\rho_n$ with the identity $\rho_n=(F_{n+2}+F_{n+1}\lambda)/(F_{n+1}+F_n\lambda)$ and the closed form for Fibonacci numbers produces the coefficient $\beta_1=\ln(2\lambda/(\lambda+2))$.
Extended reading notes
Core claim
The core claim is that at weak disorder every error acts on a state that has already relaxed to the golden-ratio fixed point, so the ratio after an error follows a deterministic sequence $\rho_n=(F_{n+2}+F_{n+1}\lambda)/(F_{n+1}+F_n\lambda)$ with $\lambda^2=\lambda+1$. Consequently the stationary ratio distribution is $P(r)=\epsilon\sum_{n\ge1}(1-\epsilon)^{n-1}\delta(r-\rho_n)$. Averaging $\ln r$ with respect to this distribution gives the Lyapunov exponent $\beta(\epsilon)=\ln\lambda+\epsilon\ln(2\lambda/(\lambda+2))+O(\epsilon^2)$. For the variant in which an erroneous rule doubles the previous element, $x_{n+1}=x_n+x_n$, the analogous formulas with $\rho_n=F_{n+2}/F_{n+1}$ are exact for all $\epsilon$, not merely asymptotic: the ratio distribution is $P(r)=\epsilon\sum_{n\ge1}(1-\epsilon)^{n-1}\delta(r-F_{n+2}/F_{n+1})$ and $\beta=\epsilon\sum_{n\ge1}(1-\epsilon)^{n-1}\ln(F_{n+2}/F_{n+1})$. The paper further shows that the support of the ratio distribution is contained in $[(1+\sqrt3)/2,\sqrt3]$ and that its cumulative distribution is a devil's staircase.
Load-bearing premise
The calculation assumes that errors are so rare that the ratio always relaxes back to the golden ratio before the next error occurs; if two errors happen close together, the first-order ratio distribution would no longer be correct.
Editorial extensions
If this is right
- At small $\epsilon$ the growth slowdown is linear in $\epsilon$ with coefficient $\ln(2\lambda/(\lambda+2))$, so the first correction to $\ln\lambda$ is fixed by the golden ratio alone.
- Typical growth is always slower than average growth, since the first-order correction to $\beta$ is more negative than that to $\gamma$; a long single run grows more slowly than the ensemble mean.
- The stationary ratio distribution is singular for every $\epsilon<1$, so the cumulative distribution is a devil's staircase and the support is a fractal set with gaps.
- In the doubling variant, the formulas for the ratio distribution and Lyapunov exponent are exact for all $\epsilon$, providing a solvable case against which perturbative and numerical methods can be tested.
Reading between the lines
- The same rare-error perturbation should apply to any one-dimensional map with a stable fixed point and an arbitrarily chosen replacement rule: the weak-disorder ratio distribution would again be a geometric sum of the deterministic post-error orbit, and the first-order Lyapunov correction would be the sum of log-deviations along that orbit.
- For the exact doubling model, one could compute higher-order terms and the variance of $\ln r$ explicitly; comparing those with analogous exact results for the average growth would show how the multiscaling spectrum builds order by order.
- Because the model is motivated by sunflower spiral counts, the empirical frequency near 95% could be treated as an estimate of $1-\epsilon$, turning the formulas into a quantitative prediction for how disordered natural phyllotaxis should affect spiral growth.
Original abstract
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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study how weak disorder affects the growth of the Fibonacci series. | contribution 0.77 | 0.87 | 0.04 | unclear 0.57 |
| s1 | 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. | method 0.68 | 0.90 | 0.04 | asserted 0.96 |
| s2 | We focus on the weak disorder limit and obtain the Lyapunov exponent, that characterizes the typical growth of the sequence elements, using perturbation theory. | method 0.75 | 0.96 | 0.28 | asserted 1.00 |
| s3 | The limiting distribution for the ratio of consecutive sequence elements is obtained as well. | result 0.99 | 0.93 | 0.13 | asserted 1.00 |
| s4 | A number of variations to the basic Fibonacci recursion including shift, doubling, and copying are considered. | contribution 0.98 | 0.88 | 0.03 | asserted 0.99 |
What was examined, and what was not
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Figures
Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603117 by the authors.
References
Discussion
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