Random feedback shift registers and the limit distribution for largest cycle lengths

Abstract For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $\mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.