Embree-Trefethen constant

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

For a fixed real β, consider the recurrence 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 one, 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.


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