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arXiv · 1903.09183

Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths

Richard Arratia·E. Rodney Canfield·Alfred W. Hales
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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 behavior was conjectured by Golomb, Welch, and Goldstein in 1959.

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Richard Arratia, E. Rodney Canfield, Alfred W. Hales. 2022-07-24. Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths. https://arxiv.org/abs/1903.09183

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