Viswanath's constant

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Number

$V$

1.1319882487943

Definition

Let $t_1=t_2=1$ and $t_n=\pm t_{n-1}\pm t_{n-2}$, each sign independent and each of the two values equally likely. Viswanath's constant $V$ is the almost-sure limit $V=\lim_{n\to\infty}|t_n|^{1/n}$ [1] [5]. The random Fibonacci sequence grows exponentially, at a rate that does not depend on the signs drawn.

Formulas

(1)

$\log V=\lim_{n\to\infty}\frac{1}{n}\log|t_n|$ almost surely, which is the Lyapunov exponent of the product of the random matrices $\begin{pmatrix}\pm1&\pm1\\1&0\end{pmatrix}$.

Comments

(2)

The constant is hard to compute: it is the Lyapunov exponent of a product of random matrices, and there is no series for it. OEIS marks the expansion hard [4] and holds fourteen significant digits; three further digits proposed in 2017 were withdrawn in 2018 as doubtful. What is stored here is what is not in doubt.

(3)

$\log V$ is the Lyapunov exponent of the random Fibonacci recurrence, and is listed with the other exponents in Lyapunov exponents of classical chaotic systems.

(4)

Oliveira and de Figueiredo computed $V$ in interval arithmetic [2], so an enclosure rather than an estimate exists in the literature; Bai later extended the expansion by a cycle expansion [3].

References

[1]

D. Viswanath, Random Fibonacci sequences and the number 1.13198824..., Mathematics of Computation 69 (2000), 1131-1155. (doi)

[2]

J. B. Oliveira and L. H. de Figueiredo, Interval computation of Viswanath's constant, Reliable Computing 8 (2002), 131-138. (doi)

[3]

Z.-Q. Bai, On the cycle expansion for the Lyapunov exponent of a product of random matrices, Journal of Physics A 40 (2007), 8315-8328. (doi)

Links

[4]

OEIS A078416: decimal expansion of Viswanath's constant

[5]

Wikipedia: Viswanath's constant

[6]

MathWorld: Random Fibonacci Sequence

Similar tables

Lyapunov exponents of classical chaotic systems — holds $\log V$, the Lyapunov exponent of the same recurrence

the golden ratio $\varphi$ — the growth rate of the Fibonacci sequence itself, where every sign is $+$; $V<\varphi$

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Table is complete: yes

How they were obtained:

Transcribed from OEIS A078416 [4], which holds fourteen significant digits and marks the expansion hard. It is written with the precision it claims under NumberDB's decimal convention, so the last digit is the uncertain one.

Not computed here, and not an enclosure: an interval computation of $V$ exists [2] and this table does not reproduce it, which is why the rigour is heuristic rather than proven. Three digits beyond the fourteen were proposed in 2017 and withdrawn in 2018 as doubtful, and are not stored.