Proof of the Lyons--White Conjecture

Abstract

Let D_n be the dihedral group of order 2n. Consider a continuous-time random walk on D_n driven by arbitrary symmetric rates whose support generates D_n. For p∈[1,∞], we say the pair (D_n,p) is rate-monotonic if for each fixed time t, the ℓ^p-distance between the random walk's distribution at time t and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that (D_n,2) and (D_n,∞) are rate-monotonic. Somewhat counterintuitively, they found several pairs (D_n,p) with p∈[1,1.997]∪[2.001,3.999]∪[4.001,5.995] that are not rate-monotonic, and they asked whether any such pairs exist with p=4 or p=6. We resolve their question, proving that (D_n,2m) is rate-monotonic for all positive integers m and n. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real p≥ 1 that is not an even integer, there exists a positive integer n such that (D_n,p) is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.

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BibTeX RIS

BibTeX

@misc{defant2026proof,
  title = {{Proof of the Lyons--White Conjecture}},
  author = {Colin Defant and Ken Ono},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27708},
  eprint = {2608.27708},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27708},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Proof of the Lyons--White Conjecture
AU  - Colin Defant
AU  - Ken Ono
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27708
UR  - https://arxiv.org/abs/2608.27708
AB  - Let D_n be the dihedral group of order 2n. Consider a continuous-time random walk on D_n driven by arbitrary symmetric rates whose support generates D_n. For p∈[1,∞], we say the pair (D_n,p) is rate-monotonic if for each fixed time t, the ℓ^p-distance between the random walk's distribution at time t and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that (D_n,2) and (D_n,∞) are rate-monotonic. Somewhat counterintuitively, they found several pairs (D_n,p) with p∈[1,1.997]∪[2.001,3.999]∪[4.001,5.995] that are not rate-monotonic, and they asked whether any such pairs exist with p=4 or p=6. We resolve their question, proving that (D_n,2m) is rate-monotonic for all positive integers m and n. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real p≥ 1 that is not an even integer, there exists a positive integer n such that (D_n,p) is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.
ER  -

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