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Routing densities and representation contraction for Thorp sweeps

OpenAI · September 26, 2026

Lean

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Abstract

For $N=2^d$ cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order $\Theta(\log N)$ in physical shuffles.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope notes for family 238 (lean/docs/238.md, 2026). States that nine routing-density and representation estimates of this paper are formalized (ThorpRouting.lean), the conditional high-height estimate requiring cutoff $J>2s$; the paper's changed later statements and six other result blocks are outside the scope. [family-level]

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Cite

@misc{OAI:Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026,
  author = {{OpenAI}},
  title = {{Routing densities and representation contraction for Thorp sweeps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf}{OAI:Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/238.md. The Lean statements have not been checked by people against the paper's claims.

Family 238: Optimal logarithmic mixing of the Thorp shuffle

Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\Theta(\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

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