Embree–Trefethen constant

In number theory, the Embree–Trefethen constant is a threshold value labelled β*.[1]

For a fixed positive number β, consider the recurrence relation

$x_{n+1}=x_n \pm \beta x_{n-1} \,$

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

$\sigma(\beta) = \lim_{n \to \infty} (|x_n|^{1/n}) \,$

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 1, and

σ > 1 for β* < β,

so they grow exponentially.

Regarding values of σ, we have:

The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.

References

1. ^ Embree, M.; Trefethen, L. N. (1999). "Growth and decay of random Fibonacci sequences". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 455 (1987): 2471. doi:10.1098/rspa.1999.0412.