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Optimal-order mixing of the Thorp shuffle

OpenAI · September 26, 2026

Lean

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Abstract

We prove that the Thorp shuffle on $2^d$ cards mixes in $\Theta(d)$ complete shuffles. The full permutation law after $1600d$ shuffles converges to uniform in total variation as $d\to\infty$, uniformly over the initial deck, while a support count gives a lower bound of $2d-O(1)$.

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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.

Follow-up works

Formalization

  • OpenAI math release, Lean scope notes for family 238 (lean/docs/238.md, 2026). States that the formalization proves full-deck total-variation convergence after $1600d$ shuffles together with the 2d−O(1) support lower bound, using the comparator files ThorpFirstReciprocal.lean and ThorpRemaining.lean. [family-level]

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Cite

@misc{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026,
  author = {{OpenAI}},
  title = {{Optimal-order mixing of the Thorp shuffle}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf}{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-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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