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

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