oai-math-revised beta

Browse › Probability and statistical mechanics › 238 Optimal logarithmic mixing of the Thorp shuffle

Random-subspace tests and trace smoothing for coordinate sweeps

OpenAI · September 26, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We prove that an absolute number of coordinate sweeps of the Thorp shuffle on $n=2^d$ positions brings the full permutation to total-variation distance at most $\frac12n^{-5}$ from uniform, uniformly over the initial deck. Thus $O(d)$ physical shuffles suffice, which is optimal in order.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

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 the formalization proves the uniform trace-smoothing estimate (regular trace at most 1+(2^d)^{-10}) and total-variation distance at most $\tfrac12(2^d)^{-5}$ after a fixed number of sweeps, for every initial deck (CoordinateTrace.lean). [family-level]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026,
  author = {{OpenAI}},
  title = {{Random-subspace tests and trace smoothing for coordinate sweeps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf}{OAI:Random-subspace-tests-and-trace-smoothing-for-coordinate-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.

Other papers in this family: