Embree-Trefethen constant

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 "-".

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

<math>\beta(\sigma) = 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.

For

0 < β < β* = 0.70258 approximately,

solutions to this recurrence decay exponentially as n→∞ with probability one, whereas for

β > β*

they grow exponentially.

Regarding values of σ, we have:

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

External link

• Proc. Roy. Soc. London Series A 455 (1999) 2471–2485