Viswanath's constant
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Viswanath's constant is a mathematical constant, occurring in number theory - more specifically in the study of randomized Fibonacci sequences. The value of Viswanath's constant is approximately 1.13198824.
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Definition
The constant is defined as the exponential rate at which the average absolute value of a random Fibonacci sequence increases. A "random Fibonacci sequence" is a sequence of numbers fn with the following recursive definition: f0 = 1, f1 = 1, and
- <math>
f_n = \left\{\begin{matrix} f_{n-1}+f_{n-2}, & \mbox{with probability 0.5}\\ f_{n-1}-f_{n-2}, & \mbox{with probability 0.5}\end{matrix}\right. <math> In other words, the decision whether to add or subtract the previous two elements of the sequence to get the next element, is taken at random with a probability of 0.5 favouring each decision (say with a toss of a fair coin.)
In a sequence, thus constructed, with a probability of 1 (i.e. with extremely rare exceptions, almost surely) the nth root of the absolute value of the nth term in the sequence converges to the value of the constant, for large values of n. In symbols,
- <math> \sqrt[n]{|f_n|} \to 1.13198824\dots \mbox{ as } n \to \infty. <math>
Explication
The constant was discovered by Divakar Viswanath in 1999 (see references). His work uses the theory of random matrix product developed by Furstenberg and Kesten, the Stern-Brocot tree, and a computer calculation using floating point arithmetics validated by an analysis of the rounding error.
Johannes Kepler had shown that for normal Fibonacci sequences (where the randomness of the sign does not occur), the ratio of the successive numbers converged to the golden mean, which is approximately 1.618. Thus, for any large n, the golden mean constant raised to the power of n yields the nth term of the sequence, with astonishing accuracy.
The random Fibonacci sequence, defined above, is the same as the normal Fibonacci sequence if always the plus sign is chosen. On the other hand, if the signs are chosen as minus-plus-plus-minus-plus-plus-..., then we get the sequence 1,1,0,1,1,0,1,1,... However, such patterns occur with probability zero in a random experiment. Surprisingly, the nth root of |fn| converges to fixed value with probability one.
Significance
In 1960, Hillel Furstenberg and Harry Kesten had shown that for a general class of random matrix products, the absolute value of the norm of product of n factors converges to a power of a fixed constant. This is a broad class of random sequence-generating processes, which includes the random Fibonacci sequence. This proof was significant in advances in laser technology and the study of glasses. The Nobel Prize for Physics in 1977 went to Philip Warren Anderson of Bell Laboratories, Sir Nevill Francis Mott of Cambridge University in England, and John Hasbrouck van Vleck of Harvard "for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems".
Viswanath's proof, by specifying the value of the constant number in this case, has helped make this area more accessible for direct study. Viswanath's constant may explain the case of rabbits randomly allowed to prey on each other. (See Fibonacci sequence for the original statement of the rabbit problem.) This step would allow closer simulation of real world scenarios in various applications.
See also
The Embree-Trefethen constant describes the behaviour of the random sequence fn = fn-1 ± βfn-2 for different values of β.
External links
- A brief explanation (http://sciencenews.org/sn_arc99/6_12_99/bob1.htm)
References
Divakar Viswanath (2000), Random Fibonacci sequences and the number 1.13198824.... Mathematics of Computation 69 (231), 1131–1155.