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2 Explication 3 Significance 4 External Links |
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 Fibonacci numbers that have the following recursive definition.
Initial conditions:
In a sequence, thus constructed, with a probability of 1 (i.e. with extremely rare exceptions, almost surely) the ratio of the absolute values of successive terms converges to the value of the constant, for large values of n.
The constant was discovered by Divakar Viswanath in 1999.
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. 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.
Though it seems surprising that a similar ratio could be obtained for a series of elements obtained by randomly chosen signs, a little thought shows that there are extremely rare cases where this ratio does not hold. For example, consider the following series
1,1,0,1,-1,0,...
This series is not allowed to "grow beyond" 1 or -1, only because the oscillating signs of + and - appear in a systematic pattern, but in a random experiment, the chances that such patterns of + and - are obtained are negligible.
In 1960, Hillel Furstenberg and Harry Kesten had shown that for a a general class of random matrix products, the absolute value of the nth term 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.Definition
Recursive step:
where,
or 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.)Explication
Significance