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

Richard Arratia; E. Rodney Canfield; Alfred W. Hales

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

Combinatorics 2022-07-26 v4

Authors: Richard Arratia , E. Rodney Canfield , Alfred W. Hales

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.

Cite

@article{arxiv.1903.09183,
  title  = {Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths},
  author = {Richard Arratia and E. Rodney Canfield and Alfred W. Hales},
  journal= {arXiv preprint arXiv:1903.09183},
  year   = {2022}
}

Comments

42 pages; 21 references, 7 sections. The cover date, July 22 2022, is hard-wired. Revised proof for Lemma 3, and a few other minor changes

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

Abstract

Cite

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