Embree-Trefethen constant

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In mathematics, the Embree-Trefethen constant is a threshold value in number theory labelled β*.

For a fixed real β, consider the recurrence

x_n+1=x_n±βx_n-1

where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "-".

It can be proven that for any choice of β, the limit

<math>\sigma(\beta) = \lim_{n \to \infty} (|x_n|^{1/n})</math>

exists almost surely. In informal words, the sequence behaves exponentially with probability one—and σ(β) can be interpreted as its almost sure rate of exponential growth.

We have

σ < 1 for 0 < β < β* = 0.70258 approximately,

so solutions to this recurrence decay exponentially as n→∞ with probability one, and

σ > 1 for β* < β,

so they grow exponentially.

Regarding values of σ, we have:

• σ(1)=1.13198824... (Viswanath's constant), and
• σ(β*)=1 .

Literature

Embree, M., and L.N. Trefethen (1999): Growth and decay of random Fibonacci sequences. Proceedings of the Royal Society London A 455(July):2471-2485de:Embree-Trefethen-Konstante it:Costante Embree-Trefethen